It depends on what parasite you are talking about. If by "contagious," you mean spread through casual contact, they can be if proper hygiene is not followed. parasites like candida and chlamydia and trichomonas are certainly contagious - spread through intercourse. Lice are contagious as are scabies.
Tapeworms and roundworms are not as easy to get from another human, but it's possible, again if proper hygiene isn't followed.