If I understand this correctly, then for p > 5 the vectors (a,0) and (0,a) in Y(2,p) should have to lift to vectors (a+kp,lp) and (mp,a+np) in X(2) which are a basis of Z^2. But this shouldn't be possible unless they form a matrix with determinant \pm 1, i.e. (a+kp)(a+np) - nmp^2 = \pm 1, In particular this requires a^2 = \pm 1 (mod p), but this is impossible for a=2.
This should also generalize in the obvious way to n > 2: we get a^n = \pm 1 (mod p), and if a is a primitive root modulo p then a^n = \pm 1 (mod p) implies a^{2n} = 1 (mod p), which can't be solved for p > 2n+1 (as well as lots of smaller p).
This doesn't eliminate all the cases that don't work, but we already see that for fixed n only finitely many p will work.