I'm not an acoustical expert, but I'll second Greg's feedback. I have a setup that is similar, but arguably not quite as bad, as your parent's. Living room open to the right to the kitchen, and to the front-right to a vaulted entry way. Living room is carpeted, but the entry way and kitches is hardwood. Ceilings are flat (except for the vaulted entry way). Leather furniture, with couch up against the back wall with some windows directly behind it.
I have Onix Reference 1s as mains, Ref 100 as a center, and a UFW-10 as a sub (still waiting on the surrounds). This is a recent upgrade from an older, two-channel system using TDL bookshelves and no sub. I can say with complete certainty that the Ref1s make for a substantial upgrade over my old system. SQ and imaging is better without question - the upgrade was well worth it. Is it as good as it could be? Obviously not, but the system still sounds better than it did.
So, no, I don't think the room will make the Axiom system sound like a cheapo setup, at least when they are both listened to in the same setting. One thing you should watch for is a spike in the bass repsonse with the main seating position up against the back wall. I get that pretty bad - I've even measured it. Good news is you can control that with the electronics and/or the sub, as long as you are willing to accept a lower bass response elsewhere in the room.
Also, I have read that Axioms are not a good match for a bright room. Perhaps Aperions or Rockets would be a better match?