Let $\overline{\mathcal{M}}(N,\mathcal{O})$ be the minimal compactification of the Hilbert modular scheme over $\mathbb{Z}_p$ for a totally real field $F$ and $\omega$ be the determinant of the co-normal sheaf ($p\nmid N$). It is known that $\omega$ is ample. We have an exact sequence of sheaf such that the first map is multiplying by $p$.

$0\rightarrow \omega^{k} \rightarrow \omega^{k} \rightarrow \omega^{k}/p\rightarrow 0$

Since $\overline{\mathcal{M}}(N,\mathcal{O})$ is projective an $\omega$ is ample, then for a large $k$, $H^{1}(\overline{\mathcal{M}}(N,\mathcal{O}), \omega^{k})$ is trivial. Moreover, the support of $\omega^{k}/p$ is the special fibre of $\overline{\mathcal{M}}(N,\mathcal{O})$, Hence by applying the global section to the above exact sequence, we find that $M_k(N,\mathbb{Z}_p) \otimes \mathbb{F}_p$ surjects to $M_k(N,\mathbb{F}_p)$.

You can jump on the weight by multiplying by the Hasse modular form.

In the case where we have a modular curve $X_1(N)$, we can use the same argument combined with Riemann-Roch to show that $H^1(X_1(N),\omega^{k})$ is trivial for a large $k$.