In previous editions, there was precedent for the remove disease spell curing parasites. For example, in 3.5e, the Red Slaad had this ability:
Implant (Ex): A red slaad that hits with a claw attack can inject an egg pellet into the opponent's body. The affected creature must succeed at a Fortitude save (DC 17) to avoid implantation. Often the slaad implants an unconscious or otherwise helpless creature (which gets no saving throw). The egg gestates for one week before hatching into a blue slaad that eats its way out, killing the host. Twenty-four hours before the egg fully matures, the victim falls extremely ill (-10 to all ability scores). A remove disease spell rids a victim of the pellet, as does a successful Heal check (DC 20) by someone with that skill. If the check fails, the healer can try again, but each attempt (successful or not) deals 1d4 points of damage to the patient.
In 5e, there is no remove disease spell, but the lesser restoration spell can remove diseases, so I'd recommend that you try that first.
Also note the use of the heal ability, which suggests that you can simply perform an operation to remove the parasite.
There's also the potential for a spell such as spirit guardians to work. This spell targets an area, and then it deals damage to any creature that "enters the area for the first time on a turn or starts its turn there". A DM might rule that parasites can be protected by their total cover, but it sounds like the spirits are incorporeal, so it's possible they can simply phase through flesh.
You might also consider moonbeam (also targets an area, though your DM is more likely to rule that cover protects against the effect) and shatter (more plausible that the sound waves can deal damage even to something inside another creature).