If he is still eating, it's not very likely that he's got internal parasites. A lot of the time when a betta spends too much time in a cup, bits of slime coat come off and the feces breaks down a lot and can look like slimy stringy bits. The best way to figure out if he has internal parasites is to put him in a bare bottomed container--I use rubbermaid/sterilite plastic storage bins. This way you can clearly see what his feces looks like.
Generally, when a fish has internal parasites they have a distended belly, they are less active or lethargic, they don't want to eat, and their poop is abnormal. If you notice more symptoms like these, I highly suggest a medication with the ingredient praziquantel. I have had some success with Jungle Parasite Clear tabs.
You should also consider changing to a higher quality pellet food brand. Hikari leaves much to be desired nutritionally. I recommend OmegaOne Betta Buffet Pellets, Atison's Betta Pellets, Ken's Betta Crumbles, and New Life Spectrum. It's also a great idea to supplement his pellet diet with a few frozen foods such as brine shrimp and blood worms. Remember to always pre-soak any dry food you feed to your betta in a bit of tank water--this helps prevent bloating.