Post-Punk evolved out of Punk, Alternative Rock evolved out of Post-Punk. Although it has taken on its own identity, Alternative Rock and Indie Rock are definitely derivative to Punk to some extent. They are essentially punk that focuses more on melodies than rhythm and riffing.
However, I don't think you'll really "get it" unless you look back on how all this stuff formed.
First off, I don't know if you'll even consider the Ramones punk by your standards, but they were the supposed originators of the genre as it was in New York. If you've never heard their stuff, here's a taste:
Their sound was based more on groove and poppy riffs, and was definitely one of the catalysts for the post-punk genre. On the other side of the pond, the Sex Pistols were originating their own style of punk, less based on pop, and more hard-hitting and anarchic:
After these two bands, many more Punk and Punk-inspired bands started popping up. There was a bit of an underground powerhouse in fact, back in the early days of punk. Because the sound of the scene was not set in stone yet, many bands began very quickly branching away from the basic foundation of punk. One of those directions was hardcore punk. After all, if we could have loud, fast, and heavy-hitting bands, why couldn't we have louder, faster, and heavier-hitting bands like these:
These bands and their sound obviously evolved into the punk sound of today, so then where does post-punk fit in?
You know how punk was really loud, heavy and fast? There were some who enjoyed the simple chord patterns and melodies, but not the fast tempos and the extreme distortion. These bands very quickly went from punkier:
and then split off in way too many directions to count:
Seriously, it's impossible to keep track. If you want to explore Post-Punk more, you can check out
this chart with nearly every significant album in the post-punk mythos.
So where did post-punk go? Well, as for itself, it pretty much disappeared after the early 80s. However, it's legacy is obvious:
Anyways, back to the topic. Why is post-punk considered punk?
The answer: It really isn't. It's just like how you wouldn't consider metal to be rock just because one evolved from the other. Post-punk evolved from punk, and ended up becoming so significant that it became its own unique "genre", before imploding on itself and spawning the music scene of the 80s and 90s.