1966 Valiant...should I part or fix it?
Less rustbucket 66's out there won't make the world a worse place. Where would you get the pieces to fix that car? A rust free body? Why not just start out with a rust free body? No, really, once "the big piece" is worn out weigh your options carefully.
It sounds like you have your plate pretty full with family, school and life in general. Cars are only transportation appliances, and old cars are just a hole in the driveway to throw money into. Kids only grow up once, and women resent the competition. I was in your situation not so long ago myself, although I parted a 66 Val and a 72 Dart, to focus on the 65 which sits behind my garage waiting for me to get the time and money to put it together. It will still be there when I get to it. Or, if it isn't, I'll find something else then.
I'm not trying to sell your stuff for you, but as a "reformed pack rat" I can tell you that launching stuff is somewhat liberating. My wife and I celebrated our 10th anniversary by getting rid of half the stuff we had, and with our 20th coming soon, I look forward to doing it again. My cheap *** figures it's free square footage, but if that doesn't work for you here's a couple others. It's better that the parts are used by somebody than if they sit rusting. Or, the more projects you have, the less chance you'll actually finish one.
Part the 66, spend most of the proceeds on the 72, and the rest on her, you'll be happy you did.