When/what to file for greencard for parents

[questionaire]

I am to be a citizen next week. After the oath ceremony, would I be able to apply for my parents' green card? My father is in US on B1 with his I-94 expiring in a month. We applied for his I-94 extention (I539) but the processing time is more than 6 months. My mother is currently not in US.
--What for do we need to file, I-130 and I-485? And I-131? (the USCIS site indicate you have to wait till a valid visa # before you can file I-485?)
--Can my dad stay (over his I-94) if we processed these (and which one) with a receipt?
--Would it be beneficiary to apply for both of them together, since my mother is not in a rush to visit? but will in a few years staying with us? If we do apply for both of them together, what is the difference for somebody who is in US vs not?

Thanks for any insight, the USCIS website is not very helpful and it's been long since I applied for myself.

[PraetorianXI]

file for the GC when they are BOTH, read to MOVE PERMANENTLY to the USA.

A Green Card is not an "enhanced" visitor visa... it is to live in the USA (not to visit every so often).

It is NEVER ok, for a visitor to overstay their visas... Look into "Consular Processing"