The first ... if not the first two ... was a parody of TTRPG tropes at the time — in particular D&D.

It went on to become a parody of fantasy literature more widely — including, naturally, LotR amongst many others (the most obvious being Cohen the Barbarian).

But it is a satire of the genre ... and a brutal one at that ... that transcends (in both in Swiftian and Carrollian manner) any and all of the others' limitations.

Satire is not parody — that for which one feels no empathy (even should it simply be a recognition of one's own, misplaced, youthful infatuation) of any kind one cannot satirise ... for one lacks the insider knowledge necessary to do so.

So, of course you will detect elements of LotR, amidst a great many other works of literature, cinema, music, art. music (fantasy, science fiction and the 'greats') — you'd have to have been born under a rock sand tayed there not to.

And, of course you will detect a certain affection (if not love) for much of it.

But, just as a patriotic person may love their country without being blind to its faults and ... as a satirist ... parody it and its people in the hope that it and they might be persuaded upon another course, so the Discworld is ... well, it is to mirrors of us, our world (and also a specific corner of it along the way), what Klaatchian coffee and knurd are to insobriety; don't mistake its gentle nods to its antecedents as even stanish fervour, never mind unthinking idolatry — you might love your daughter, but that doesn't mean you aren't critical of her at the same time ... I might smile wryly when mention is made of a band I idolised in my youth, but I'd have no difficulty parodying/satirising them and then sleeping well that night (even without the prospect of there being money in my bank account as a result)

Yes, you have detected elements in his work of ... wait for it ... the very things he was parodying/satrising — give yourself a pat on the back (there's a gold star waiting for you at the end of the school day). But, you know what ... I'd hope so — it's a poor satirist who leaves you wondering who (or what) the target of their ire is, after all ... and a poor reader who cannot discern it.

Wait until you've read all the Discworld novels — we can discuss your thoughts on the matter in light of your new experiences then.