The Proto-Indo-European root *péleḱu- 'axe' seems to me to be contrived. While the correct centum-satem correspondence between Greek -k- and Sanskrit -ś- is reassuring, it by no means validates the reconstruction. First, we have two fullgrade vowels in two consecutive syllables which immediately gives the alleged root an un-Indo-European appearance. Second, the root rests solely on the strength of a comparison between just two cognates: Greek πέλεκυς pélekus and Sanskrit परशु paraśú-. Third, their respective word accents don't match.
On the other hand, there seems to be a controversy concerning the skeptic's attempt to relate the above lexemes with Assyrian pilaqqu which has been long claimed to mean 'axe' but which, given the new reading of 'spindle', is thereby discredited. Ironically however there remains Assyrian palāqu 'to fell, to slaughter' to contend with, a verb on which nominal derivatives like naplaqtu 'knife' were built. Surely this is fundamentally not an Indo-European word and a Semitic source like this one remains preferable over a PIE root that begs even more questions than it's worth.
Yet, while we're on the subject. Are we sure that Sanskrit परशु paraśú- is really connected to the Greek? Consulting the online Monier-Williams Sanskrit-English Dictionary, an alternative form, पर्शु parśú-, is also provided. Unless I'm overlooking something, this only adds further doubt to the comparison. I wonder in what way any IEist can provide reassurance that these two words should be related. Note also Akkadian parāsu 'cut off, cut into pieces, separate' which should warn us that, considering many competing sources of this word available, these cognates could be red herrings. It would help greatly if we could at least establish the antiquity of this word in Proto-Indo-Iranian by way of further evidence.