Who are Linus' mom and dad really?

First of all, I'd like to point out that the Ocean's movies are repeatedly confronting you with various references to persons and events which are only mentioned in passing and whose backstory is only hinted at slightly, but whose meaning and backstory is completely clear to everybody else except you as the audience. This is IMHO part of the movies' classy and self-confident style and helps to establish this kind of parallel world of thieves and robbers we're presented with. And especially Linus's parents are introduced as some kind of legends that everybody else seems to know but whose actual nature stays a mystery to the audience. So I think there is not much of a backstory one can take directly from the movies and this is by-design.

What I concluded from the movies was, that Linus's parents are in the same criminal business as all the others and seem to be pretty well-known and respected in this business. This is also slightly hinted at by Linus's refusing any help from his parents, which suggests he's somehow treading into the same carreer path as them but wants to prove his value by doing things on his own. So those scenes in Ocean's 12 and Ocean's 13 where they appear as "official government people", are actually just big frauds in order to get Linus out of trouble (and give you that usual wow-effect of "wait, that was all just a staging?").

We are actually provided a quick backstory in Ocean's Eleven when Linus is caught by Danny after picking someone's pocket.

Hello, Linus. Whose is this?

Who are you?

A friend of Bobby Caldwell's.

You're either in or out. Right now.

What is it?

It's a plane ticket. A job offer.

You're pretty trusting.

Bobby has faith in you.

Fathers are like that.

He didn't tell you? He doesn't want me trading on his name.

Do this job, he'll be trading on yours. Don't, and you can go back to feeling up stockbrokers.

Can you get the check, please?

That's the best lift you've made yet.

Going by this conversation, we can infer Linus' dad is Bobby Caldwell who is also a con. His mother is very likely one too.

There's another conversation between Reuben and Linus where the latter is referred to as "Bobby Caldwell's kid".

Yep, they say that Linus' dad has the best cover ever, meaning that probably his parents (or his dad only) are really part of the FBI while doing their business (no wonder no one ever caught them).

He als says when linus is on the phone with his dad in oceans 13 that he has te best cover for this buissines:s so he probably has a good job or something