Yes, parents do count as immediate family.
That said, note that the family privilege only applies if you're TK Elite or Elite+. It doesn't apply for other *G members, who are still allowed to bring only one guest.
Re: paid access - no, there's no paid access at the IST lounge.
Special entrance wise, yes there's a priority passport control queue that leads directly into the lounge, but bear in mind that again you'll have to be either TK's own gold or fly in J/F to be able to use it. *G flying in Y can use the "normal" priority line, which is a few booths further back. Also, I can't confirm this but I doubt that priority passport control would apply to your whole family.