What exactly is rebirth? And what exactly is death?
Most people take rebirth to be the rebirth of oneself into a new life. If we zoom further into what this means, this is the continuation of the same consciousness that is aware of its surroundings and its thoughts into a new body with a new identity and new life. The same consciousness from birth wandered through life till death, then it continues in a new life after rebirth.
The Buddha did not accept this view in MN 38:
The Blessed One then asked him: “Sāti, is it true that the following
pernicious view has arisen in you: ‘As I understand the Dhamma taught
by the Blessed One, it is this same consciousness that runs and
wanders through the round of rebirths, not another’?”
“Exactly so, venerable sir. As I understand the Dhamma taught by the
Blessed One, it is this same consciousness that runs and wanders
through the round of rebirths, not another.”
“What is that consciousness, Sāti?”
“Venerable sir, it is that which speaks and feels and experiences here
and there the result of good and bad actions.”
“Misguided man, to whom have you ever known me to teach the Dhamma in
that way? Misguided man, have I not stated in many ways consciousness
to be dependently arisen, since without a condition there is no
origination of consciousness? But you, misguided man, have
misrepresented us by your wrong grasp and injured yourself and stored
up much demerit; for this will lead to your harm and suffering for a
long time.”
The Buddha taught consciousness to be dependently originated.
This is elaborated further in the same sutta:
“Bhikkhus, consciousness is reckoned by the particular condition
dependent upon which it arises. When consciousness arises dependent on
the eye and forms, it is reckoned as eye-consciousness; when
consciousness arises dependent on the ear and sounds, it is reckoned
as ear-consciousness; when consciousness arises dependent on the nose
and odours, it is reckoned as nose-consciousness; when consciousness
arises dependent on the tongue and flavours, it is reckoned as
tongue-consciousness; when consciousness arises dependent on the body
and tangibles, it is reckoned as body-consciousness; when
consciousness arises dependent on the mind and mind-objects, it is
reckoned as mind-consciousness. Just as fire is reckoned by the
particular condition dependent on which it burns—when fire burns
dependent on logs, it is reckoned as a log fire; when fire burns
dependent on faggots, it is reckoned as a faggot fire; when fire burns
dependent on grass, it is reckoned as a grass fire; when fire burns
dependent on cowdung, it is reckoned as a cowdung fire; when fire
burns dependent on chaff, it is reckoned as a chaff fire; when fire
burns dependent on rubbish, it is reckoned as a rubbish fire—so too,
consciousness is reckoned by the particular condition dependent on
which it arises. When consciousness arises dependent on the eye and
forms, it is reckoned as eye-consciousness…when consciousness arises
dependent on the mind and mind-objects, it is reckoned as
mind-consciousness.
Please also see this answer in which Damith explained:
Think of a stream of water. When you look at a single point in the
stream, you see many water droplets appear, exist, and disappear. Take
a single droplet out of it which passed by that point. If you
consider the past, present, and future of that droplet, you'll see a
stream. In reality, it's not the same droplet nor another droplet you
see in the past, present, and future. Stream of consciousness is
analogous to this example.
So MN 38 is a rejection of consciousness as being the eternal unchanging self that moves from life to life. It's not a rejection of the continuity of the stream of consciousness that is conditioned and changing.
So, is there such a thing as rebirth? Yes. But what is it?
It's not the rebirth of oneself, but it's the rebirth of one's self - not one's soul, but one's mental idea of a self (the idea 'I am the thinker' of Snp 4.14). It's not the rebirth of the individual, but it's the rebirth of individuality.
In SN 15.3 (below), the Buddha looks at YOU (the mental idea of self) and tells YOU that YOU have been reborn so many times and grieved the death of so many loved ones (which are more mental ideas classified relative to the mental idea of the self - see this question on papanca), that the volume of tears shed by YOU (the self) is more than the volume of water in all the oceans of the world combined. Also, a beginning to this samsara is inconstruable.
Every time this mental idea of a self is reborn, the situation of rebirth is based on past karma but fueled by craving and clinging. Thoughts, words and actions motivated by clinging and burning with passion, aversion and delusion, will result in negative repercussions, and vice versa. From moment to moment, rebirth of the mental idea of the self is fueled by craving, but the situation of rebirth is based on karma.
"Long have you (repeatedly) experienced the death of a mother...
father... the death of a brother... the death of a sister... the death
of a son... the death of a daughter... loss with regard to
relatives... loss with regard to wealth... loss with regard to
disease. The tears you have shed over loss with regard to disease
while transmigrating & wandering this long, long time — crying &
weeping from being joined with what is displeasing, being separated
from what is pleasing — are greater than the water in the four great
oceans.
What is death? Of course, it is also the death of YOU, a self identity.
Then what is Parinibbana?
Well, there is such a thing as clinging aggregates. According to this answer, for a living arahant, when ignorance is uprooted, this breaks dependent origination, that ends craving, clinging and suffering. In the chain of dependent origination, clinging aggregates would also cease.
The living arahant according to Iti 44, attained nibbana with fuel remaining (sa-upadisesa), meaning the non-clinging aggregates are still functioning like glowing embers, although the fires of passion, aversion and delusion have ceased.
Parinibbana is when the non-clinging aggregates stop functioning. This is nibbana without fuel remaining (anupadisesa) according to Iti 44.
What is difference between parinibbana and any other death? In the former, the fires of passion, aversion and delusion have ceased. In the latter, the fires of passion, aversion and delusion keeps burning, the (mental idea of a) self is reborn and suffering continues.
A change of perspective is needed i.e. rebirth, birth, death, and parinibbana should not be linked to any permanent consciousness or identity. The first noble truth is that there is suffering. It's not there is my suffering or your suffering but simply there is suffering. This also relates to anatta (not self).
Also, think about the core teaching, sabbe dhamma anatta - all phenomena is not self. There is no permanent consciousness or identity or individuality or self in all phenomena.