The Japanese word for one’s parents is 両親 [りょうしん], which literally means “both parents.” If you want to refer to someone else’s parents, just slap a 御 [ご] honorific prefix on the front of it: 御両親.
Your father is 父 [ちち]. Someone else’s father is お父さん [おとうさん]. Your mother is 母 [はは]. Someone else’s mother is お母さん [おかあさん]. You can, of course, call your own mother and father the more polite version, but it would be bad form to call someone else’s father 父 [ちち]. If you want to up the politeness a bit, you can use the 様 [さま] suffix instead of さん.
Here are some other permutations:
- 父ちゃん [とうちゃん] ・ 母ちゃん [かあちゃん]
- パパ ・ ママ
- 父親 [ちちおや] ・ 母親 [ははおや]
If, for some reason, you should find yourself in pre-Meiji Japan, you might want to use 父上 [ちちうえ] and 母上 [ははうえ] to fit in. I believe this can also be used when writing very formally in reference to someone else’s parent. One of my favorite research websites, SpaceALC presents the following example:
母上がお亡くなりになったとのこと。お悔やみ申し上げます。 Please accept my sincere condolences on the loss of your mother.