Does Fighter have parents?
 

Black Mage has mentioned his siblings in the past tense a few times, Red Mage's parents showed up in that (imaginary) flashback, and Thief's dad made an appearance, but I can't remember if Fighter's family has ever shown up.

When a mommy fighter and daddy fighter really love each other...

That would be Vargus the Swordman. It's not Fighter's father.

Well, a friend of mine disagrees with me on this, but I kind of supposed that he has a sister at least, from 575.

Black Mage says that a Dire Elk bit his sister once, and Fighter says "You have a sister too?" . Since no one else was talking about having a sister in that comic, I supposed Fighter was talking about himself. Don't know about his parents, though, sorry.

Interesting comment your friend made, but inaccurate.

Black Mage previously mentioned in a previous comic and pointed out by RM that he had a brother, which he blinded and pushed down a pit, likely out of rivalry for his mom's affection. He never mentioned his sister until the comic you pointed out. That's when Fighter mentioned "You have a sister too?". Not out of Fighter having a sibling (although it doesn't rule out the possibility), but out of learning additional info about BM's family.

If I answer "no" every biologist that is currently alive will beat me with a stick.

Of all the kinds of accuracy you can't assume in 8-Bit Theater I think biological accuracy is one of the ones you most can't assume.

Whale vulture!

Does Fighter have parents?

Probably. But we've never seen them, and all his flashbacks and such generally go to his training as a swordsman or his time in Fighter's Camp. Plus, he's Fighter.
I think it's safe to assume Fighter was born, and then his parents dumped him into Fighter's Camp to get him away before it was too late for them.