In most traditions of Buddhism there is talk of the Buddha, or various Bodhisattvas talking about their past lives.
At what point in the path to enlightenment is the ability to remember ones past lives gained?
Powers are generally orthogonal to insight.
Most enlightened people have some powers, but not all people with powers are enlightened.
The six higher powers (chalabhiññā) include the cessation of ignorance, leading to becoming an arhat and the knowledge of past lives.
Thus theoretically one can be an arhat but not have knowledge of past lives, if one isn't interested in pursuing such skills or vice versa. However, this is contentious, since popular lore expects arhats to walk through walls or be able to perform other special deeds, but the Buddha himself never solely equates powers with enlightenment.
Devadutta, the villainous cousin of the Buddha had lots of powers, including transforming himself into other beings through jhana cultivation, but he was no arhat even though he was only a short hop away from it - he simply had no interest in enlightenment. He was interested instead in profiting from the psychic and political powers of being a spiritual guru.
"Remember one's former abodes" (pubbe-nivāsanussati), that is, recalling ones own past lives;
"Extinction of mental intoxicants" (āsavakkhaya), upon which arahantship follows.
"Are powers strictly necessary for enlightenment?" is a separate question. While they can be a distraction keeping one away from enlightenment, after enlightenment they can become useful tools. For example, those on the Bodhisattva path are expected at some point to focus their training on attaining mind reading powers so that they may help others to liberation.
Yet, adherents of the Zen tradition would say enlightenment is very ordinary, it was never not there. Enlightenment is doing ordinary things in a most extraordinary way.
Before enlightenment, chop wood, carry water. After enlightenment, chop wood, carry water.
Watch "Ten Oxherding Pictures" on YouTube - https://youtu.be/bN126j4FH7g
It is true that one can gain the ability (Iddhividha-niddesa) to recall past lives.
The Visuddhimagga has a large section on how to develop these supernatural powers.
In here they go into great detail on how one can develop them. There is a section on "Recollection of Past Lives" (p. 404-415), but it's too large to quote. Instead here is a quote from the introduction chapter, p. 369.
THE SUPERNORMAL POWERS (Iddhividha-niddesa)
[THE BENEFITS OF CONCENTRATION (CONTINUED)]
It was said above with reference to the mundane kinds of direct- knowledge that this development of concentration “provides ... the benefit of the kinds of direct-knowledge” (XI.122). Now, in order to perfect those kinds of direct-knowledge the task must be undertaken by a meditator who has reached the fourth jhána in the earth kasióa, and so on. And in doing this, not only will this development of concentration have provided benefits in this way, it will also have become more advanced; and when he thus possesses concentration so developed as to have both provided benefits and become more advanced, he will then more easily perfect the development of understanding. So meanwhile we shall deal with the explanation of the kinds of direct-knowledge now.
In order to show the benefits of developing concentration to clansmen whose concentration has reached the fourth jhána, and in order to teach progressively refined Dhamma, five kinds of mundane direct-knowledge have been described by the Blessed One.
They are: (1) the kinds of supernormal power, described in the way beginning, “When his concentrated mind is thus purified, bright, unblemished, rid of defilement, and has become malleable, wieldy, steady, and attained to imperturbability,1 he directs, he inclines, his mind to the kinds of supernormal power. He wields the various kinds of supernormal power. Having been one, he becomes many ...” (D I 77); (2) the knowledge of the divine ear element; (3) the knowledge of penetration of minds; (4) the knowledge of recollection of past lives; and (5) the knowledge of the passing away and reappearance of beings.
Another way to recall past lives (only one life though) is to remain conscious throughout the death-process. If you are fully conscious throughout the entire process you will become liberated. If there is still some clinging left you will be reborn as a Brahma or Deva and be able to recall fully the last life.
I asked a question on exactly that and Ven. Yuttadhammo gave this answer to my question. You can read the question and the answer here.
I remember once in a talk by Ajahn Brahms he said that a lot of his monks would recall past life memories as they were coming out of dhyana states...