It's a constant struggle for frugal musicians to afford pricey instruments that are worthy of their art. Don't worry. You're not alone.

Whether to get your parents to spend the extra buck$ is between you and your parents. So I'll stay out of that one.

Between the G2 and the C2, I think the C2 has what's called a "duplex scale" that and the G2 does not. But "duplex scale" is not necessarily a "good thing." It all comes down to whether you like the sound. So, if you want to compare, go listen to a C2 and see if you like the sound better. C1 is just smaller than the C2, otherwise, quite similar.
Exploring a new C1 is OK, just make sure you play it first and confirm for yourself that it is something you deem acceptable. Yamaha also has the GC1 which is the same size as the C1, but cheaper with some differences in design and material used. Try that too, especially if the budget is really tight. Personally, I have no problem with the C1/GC1's action, it's tone that I'd suggest that you pay attentoin too. When you go from a 5'7" to a 5'3", the bass would usually get weaker (mid-range and treble should be OK). So pay attention to the bass when you try out the C1/GC1. If you find the bass acceptable, then all is well.
At the level you're playing, you do not need any Internet stranger to tell you which piano you should buy -- you should have the confidence that you'd know quite instinctively whether a piano will be "good enough" for you once you spend 20~30 minutes playing on it seriously (hopefully without the salesperson keep interrupting). So what's mentioned below is just a starting point -- you really need to
play the piano before you buy.
Even assuming a small budget (sub-$10k), a few things you can keep your eyes out for:[list=1]
- Used "recent vintage" Yamaha C2, Kawai RX-2 or Kawai KG-2 or larger (shoot for something less than, say, 10~15 years old); If you get up to a C3, so much the better
- Used Baldwin grand (perhaps model L or larger)
With used pianos, condition is king -- first, trust you own ears and your own fingers, do not buy one without playing it first. Chances are good that you might find sellers who tell you "the piano will get better after it gets tuned/regulated/voiced/whatever" -- do not take their words for it, let them do the tuning/regulation/voicing/whatever and then you go play it again before deciding whether the piano is "good enough" for you.
The key is to resist the temptation to buy a piano you instinctively know to be "not good enough" just because it has a low price tag, and be patient.
Then, after you've found a used piano you really like, just to be careful, I'd say spend a little money to hire an independent piano technician to help you examine the piano before you buy ("independent" as in one with no conflict of interest with the seller; your current technician may be able to help). The concept is similar to getting an independent auto mechanic to help you check out a used car before you buy.
Forget about the European/Eastern European pianos. Unless you're going to spend $18k~$20k and up, I just don't see them as being practical for you. It will be an entirely separate "pros and cons" discussion if your budget do go up that high.
Good luck.