In the first picture, despite being a crown tail, it does look like he has fin rot. Here's a healthy red, crown tail betta for comparison (broken link removed)
However, in the last couple of pictures, his fins are looking better. I'm guessing having clean water has really helped.
Like others members have touched on, it is probably swim bladder. Just feed him a thawed, de shelled frozen pea chopped up into bite sized pieces.
I second getting rid of the water tablets, and getting a bottle of Prime instead. Prime is amazing. "It removes chlorine, chloramine and ammonia from tap water, and detoxifies nitrite and nitrate. It also promotes the production and regeneration of the natural slime coat of fish. Use Prime when setting up your aquarium, and whenever adding or changing water." Until then, at least get some AquaSafe, Tetra BettaSafe, or Stress Coat. You want an actual liquid conditioner, not just dissolvable tablets.
While I agree that bettas need heated, filtered tanks, filtering a tank as small as 2 gallons is unfortunately a bad idea. The filter will take up a good deal of swimming space, and unless you get an adjustable flow filter, the flow will be too high and push your betta around. Heat is essential though, as bettas need it to be around 80F or so. A cold betta, is sick and lethargic.
Since filters are out of the question, that pretty much means you have to do 100% water changes daily. This can also be stressful for fish.
Also, I would highly recommend switching out those plastic plants for some silk ones. Plastic plants are notorious for bettas tearing their fins on the sharp edges. Silk plants don't carry that risk. You can get a pack of 3 for really cheap at Walmart or any local pet/fish store.
You mention your parents are not wanting you to switch to a bigger tank. Have you tried explaining to them how essential it is? A 5 gallon won't take up much more room than a 2 gallon (although, with all that gravel in there, it's more like a 1.5 gallon).
Here's a 5 gallon that is only $30
https://www.walmart.com/ip/Hawkeye-...-with-LED-Lighting-Fish-Aquatic-Pets/17480272 You would just need a heater.
https://www.walmart.com/ip/Tetra-Heater-5-15-Fish-Aquatic-Pets/10291808 This one works well for smaller tanks, and is pretty small in size.
You could probably find a cheap 5 gallon set up for only a few dollars on craigslist as well.
I second getting a liquid ammonia kit. Read up on fish cycling if you haven't already.
Another thing, I know how hard it is to see bettas suffering and want to "rescue" them and bring them home. However, for this betta that you saved, another betta will be replacing him, kept in the same, awful conditions. It's an endless cruel cycle. The only way to break it is by refusing to purchase bettas from stores. Next time, if you really want a betta, look on craigslist and similar sites.