Any form of existence after parinirvana is "an inconceivable" in the Buddha's Dharma. This is equally true of Theravada as it is of Mahayana here. The Buddha, "after death," is inconceivable. The Buddha's mind, after death, is inconceivable. The Buddha's mind, before death, is also inconceivable. The Buddha questions Vesālī:
“What do you think, Anurādha? Do you regard the Realized One as form?”
“No, sir.”
“Do you regard the Realized One as feeling … perception … choices …
consciousness?”
“No, sir.”
“What do you think, Anurādha? Do you regard the Realized One as in
form?”
“No, sir.”
“Or do you regard the Realized One as distinct from form?”
“No, sir.”
“Do you regard the Realized One as in feeling … or distinct from
feeling … as in perception … or distinct from perception … as in
choices … or distinct from choices … as in consciousness … or as distinct from consciousness?”
“No, sir.”
“What do you think, Anurādha? Do you regard the Realized One as
possessing form, feeling, perception, choices, and consciousness?”
“No, sir.”
“What do you think, Anurādha? Do you regard the Realized One as one
who is without form, feeling, perception, choices, and consciousness?”
“No, sir.”
(Anurādhasutta SN 44.2)
The Buddha is not defined via five aggregates like a sattva is. This is before even parinirvāṇa. After parinirvāṇa, the Buddha is also not defined via five aggregates. With this in mind, we can go to the 25th Parīkṣā of the Mūlamadhyamakakārikā with the Madhyamakaśāstra commentary from ~2-400 AD by Venerable Vimalākṣa (青目, an alternate Sanskritization is "Piṅgala" )
25:17 The Tathagata, after parinirvāṇa, is neither said to exist nor not to exist, nor is he said to both exist and not exist, nor neither exist nor not exist.
25:18 The Tathagata, presently, is neither said to exist nor not to exist, nor is he said to both exist and not exist, nor neither exist nor not exist.
"Presently" here should be taken as "while alive."
25:19 Between nirvāṇa and the world, there is not the slightest differentiation. Between the world and nirvāṇa, there is also not the slightest differentiation.
25:20 From the true apex of nirvāṇa to the apex of the world, like this, there are two apices with not a sliver of difference between them.
25:21 "Existence or nonexistence after parinirvāṇa," et cetera, "the boundaries of existence," "constancy," et cetera -- all (of these) views depend on nirvāṇa being in the future or in the past.
Venerable Vimalākṣa comments:
(1) "The Tathagata after parinirvāṇa exists," or (2) "The Tathagata does not exist," or (3) "The Tathagata both exists and does not exist," or (4) "The Tathagata neither exists nor not exists," or (5) "The world is finite," or (6) "The world is infinite," or (7) "The world is both finite and infinite," or (8) "The world is neither finite nor infinite," or (9) "The world is constant," or (10) "The world is inconstant," or (11) "The world is both constant and inconstant," or (12) "The world is neither constant nor inconstant" -- these in three kinds are twelve views. When after the Tathagata's parinirvāṇa he is (1) existent or (2) nonexistent, et cetera (3-4), those four views arise dependent on (the notion of) nirvāṇa. When the world is (5) finite or (6) infinite, et cetera (7-8), those four views arise dependent on (the notion of) the future. When the world is (9) constant or (10) inconstant, et cetera (11-12), those four views are dependent on (the notion of) the past. Whether the Tathagata after parinirvāṇa exists or does not exist, et cetera, is inconceivable, and nirvāṇa is also the same. Like the origin and destiny of the world, if it is finite, infinite, constant, inconstant, et cetera, it is inconceivable, and nirvāṇa is also the same. Consequently, we say that the world and nirvāṇa are not different.
Āyuṣmat Nāgārjuna's Kārikā continues.
25:22 If all phenomena are empty, what is finite, infinite, both finite and infinite, or neither finite nor infinite?
25:23 Why is there the one and the many? Why is there the constant, and inconstant, and both constant and inconstant, and neither constant nor inconstant?
25:24 All phenomena being inconceivable is the cessation of the frivolous ponderings. To no person and in no place has the Buddha ever spoken.
Venerable Vimalākṣa comments again:
All phenomena at all times of every variety conform to
dependent origination. All in all, they are empty, and thus they have
not their own natures (自性, svabhāva). Which body exists to be the same
as the soul? Which body exists to be different than the soul? Thus
proceeding are the sixty-two demonic views. Each and every one within
emptiness is untenable. When all existence completely ceases,
frivolous ponderings are entirely gone. When frivolous ponderings are
entirely gone, it is because we have penetrated the true aspect of the
many phenomena and attained the tranquil path. Hearkening back to the
chapter on causality, if we inquire into the many phenomena, they are
neither existent nor nonexistent, nor are they both existent and
nonexistent, and nor are they neither existent nor nonexistent. This
is called 'the true aspect of all phenomena.' It is also called 'the
true nature of the phenomena,' 'reality,' and 'nirvāṇa.' Therefore,
the Tathagata at no time, in no place, to no persons, ever spoke of
nirvāṇa as with particularized characteristics. When all existence has
entirely come to an end, frivolous ponderings have ceased.
(Madhyamakaśāstra T1564.34c14)
Which directs us back to the OP inquiry: What happens to consciousness when entering parinirvāṇa? It ceases.
Of all forms of disease, none is greater than that of having a body —
he extinguished the body. Of all forms of torment, none is more severe
than that of having a calculating mind — he erased it and submerged in
the vacuous. The mind is taxed by the body; the body is burdened by
the intellect. The two pull each other, turning like a wheel on the
endless road of misery. It is said in the sutra, “The intellect is
poison, the body is shackles. Because of them the quiet silence of
liberation remains beyond reach; they are the cause of all
tribulations.”
The Perfect turned his body into ashes and extinguished his intellect,
he relinquished his form and discarded his reason. Within, he
abandoned the stirrings of illumination; without, he put to rest the
basis of misery. Transcendent, perfectly free from all existents;
boundless, he became great and vacuous. Tranquil, inaudible, clear,
non-manifest, mysteriously gone forever into a destination unknown.
When a lamp goes out its flame is extinguished, the oil and the flame
gone all at once.
This is nirvana without remainder. In the words of a sutra, “The five
aggregates are no more, like a flame extinguished.”
[...]
The terms “with remainder” and “without remainder” are only external
appellations for nirvana, conventional designations for the divergent
modes of the sages’ responding to things.
[...]
As for these (two) appellations (of "with remainder" and "without remainder"), they are established to indicate the
various modes of sagely response. When the Sage manifests traces, we
call this “arising”; when he makes them vanish, we call this
“cessation.” His “arising” is referred to as nirvana “with remainder,”
his “cessation” as nirvana “without remainder.” All along both
appellations — “with” and “without remainder” — remain rooted in the
nameless. Surely the nameless will take any name? Thus, the Perfect
becomes a square when he inhabits a square, a circle when he stops in
a circle, a deva when among devas, a human being when among humans. To
become a deva or a human being in accordance with circumstance is
surely not something that devas or humans could do. It is precisely
because he is neither a deva nor a human that he can become one or the
other.
(Venerable Sēngzhào Zhào's Essays T1858, translator unknown by poster, published by BDK as Three Short Treatises by Vasubandhu, Sengzhao, and Zongmi)
Though the Buddha's mind ended upon parinirvāṇa, he was never identified as, with, or in his mind to begin with. When I say "to begin with," I mean "after the bodhimaṇḍa," after he touched the earth in response to Mara, the godling prince. Nothing changes for the Tathāgata when form, consciousness, etc., ends, because he has already discarded, transcended, and escaped all of these.
In the Sarvāstivādin Mahāvibhāṣa ("The Great Commentary" of the Sarvāstivādin Śrāvakas), it explains nirvāṇa using some clever wordplay: "Vāna" means forest and "nir" means escape.
As it is the escape from the forest of the aggregates,
it is called nirvana.
When we ask a question like "Is the Buddha aware after parinirvāṇa," we are trying to pin the Tathāgata down, to identify him, using the aggregates by which ignorant sentient beings are bound. This can't be done.
Overlong, but hopefully helpful in some way.
