Should I see a specialist for parotitis, or can my regular doctor take care of it?

Depends. If it reallyis parotitis, one needs to hydrate agresdively, use a warm compress to the area, take sour things 3-4 times a day, massage the gland repeatedly and finally take antibiotics. If your primary md can direct you to do the above and rx antibiotics, then great. If not, one should see an otolarynhologist (ent).
Regular doctor. Most cases can be effectively managed by your regular doctor. Especially challenging cases may need the assistance of an ENT surgeon. Talk with your doctor to find out more.