For the record, I hate the abuse of laryngeals. What I mean by "abuse" is when people, unsatisfied with a protolanguage proven to contain seemingly exotic laryngeals with accompanying vocalic effects, decide to add laryngeals to every stem to account for all long vowels, whether it can be justified or not, and end up succeeding only in muddling the whole grammatical system in the process, obscuring the very thing they attempt to clarify.
A quick and easy example of this is Bhadriraju Krishnamurti's use of laryngeals in the 1st and second pronouns *yān 'I' and *nīn 'you' (or in his view, *yaHn and *niHn) to account for lengthening in the nominative which opposes oblique stems *yan- and *nin- lacking added vocalic length. Without laryngeals, it should be already clear to a knowledgeable linguist that many languages simplify stems in oblique case forms without the need to appeal to arcane infixation of one 'miracle phoneme' or another. We see this same simplification of pronominal stems between noun cases in PIE when *tu with back high vowel in the default nominative case opposes *twe. What was once *u throughout the entire 2ps pronominal paradigm must have apparently weakened at some point to a lower vowel *e in oblique cases. The phenomena exhibited in these two protolanguages are surely one and the same and therefore do not require laryngeals to explain them.
So to the topic now, there's a succinct reason why the 2nd person singular perfect ending *-th₂e contains a laryngeal that apparently even many IEists, so seduced by their own ideas, forget to take into account. It's given this laryngeal because of the Sanskrit reflex -tha whose added aspiration should otherwise not be present. As a result, we obtain a nice symmetry on the PIE level between 1ps perfect *-h₂e and 2ps perfect *-th₂e.
Yet further, it's precisely that very phonemic symmetry embedded in the perfective system that ultimately helps to explain the emergence of laryngeals in 2ps endings and stems at all! It can be readily seen from this pattern that the true reason for the laryngeal in the 2ps perfect is analogical levelling with the adjacent 1ps ending. While the laryngeal of the 1ps is legitimately ancient and stemming from the earliest stages of Pre-IE, the 2ps laryngeal cannot be. It should also be noted that without a laryngeal in this ending, it would otherwise be identical with the corresponding 2nd person plural ending *-te, adding further motivation to this irregular change.
Coincidently, there is no other evidence for laryngeals elsewhere in second person morphemes, whether it be in *tu (2ps. nom.), *twe (2ps.acc.), *tene (2ps.gen.) or *-s(i) (2ps.dur.). Putting laryngeals in places they needn't be is a grave misanalysis on the part of a comparative linguist who's obligated by Logic to find the simplest solutions possible given the available evidence.
 Krishnamurti/Emeneau, Comparative Dravidian linguistics - Current perspectives (2001), p.336.
 I should add that my theories on syllabic structure in Mid IE (ie. the stage before stress-motivated Syncope) forbid onset clustering at all via Occam's Razor. No Semitic loans identified during this period seem to require tautosyllabic consonant clustering in Mid IE either. All clusters in PIE seem to be the result of later Syncope and the changes in legal syllabic structure that ensued. Yet if we're obligated to side with this typological simplicity of Mid IE, this is yet one more reason why the onset cluster seen in *-th₂e cannot be any older than the early Late IE period. We may then presume the singular form to have been *-te before this time.