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So, assume $f$ is flat, then if $X$ is smooth, then $\Omega_X$ is still locally free and so it is flat over $Y$ and hence you get that $\dim H^q(X_y, (\Omega^p_X)_y)$ is semi-continuous. I think this is the best you can hope for, but at least this This works in any characteristic and it does not have anything to do with Hodge theory. For singular fibers $\Omega_{X_y}$The shortcoming of this is that you're actually not getting $\dim H^q(X_y, \Omega^p_{X_y})$ even in the "right" thing to look atsmooth case, so because for that you would need $\Omega_{X/Y}$, but that's not flat and in some sense it not the right object to consider.

If $Y$ is actually also smooth, then $\Omega_Y$ is locally free and if $f$ is dominant, then you still have the short exact sequence0\to f^*\Omega_Y\to \Omega_X\to \Omega_{X/Y}\to 0The trouble is that the sheaf on the right is not a big deal locally free, so when you take exterior products, then it gets kind of tricky. One possibility is to construct complexes that that's behave very similarly to $\Omega_{X/Y}^p$ with respect to $\Omega_X^p$ and $\Omega_Y^p$. This is done in this paper. The primary goal of the paper is not what you want and I am not sure that you actually get semi-continuity, but you can at least check out the construction.A general version of that construction is in this paper. (Sorry, neither of them are on arXiv).

Also, for singular fibers $\Omega_{X_y}$ is not the function"right" thing to look at.One can actually look at the objects that come from the Deligne-Du Bois complex and do Hodge theory for singular varieties. Then the restriction becomes a little tricky, because those are objects in a derived category and restriction is not exact, so you need to do something else. Completion along the fiber gives the right thing, but I don't know if there is a semi-continuity theorem using completion instead of restriction. That might be an interesting question to contemplate. This is related to the paper I linked above. See the references in that for more details.
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Without flatness you have little chance for this to even get off the ground: Let $f:X\to Y$ be the blow up of a smooth (closed) point of $Y$. Then all fibers except one consist of a single point, while the special fiber is a $\mathbb P^n$. That will have non-zero Hodge numbers that the others can't even dream about.

So, assume $f$ is flat, then if $X$ is smooth, then $\Omega_X$ is still locally free and so it is flat over $Y$ and hence you get that $\dim H^q(X_y, (\Omega^p_X)_y)$ is semi-continuous. I think this is the best you can hope for, but at least this works in any characteristic and it does not have anything to do with Hodge theory. For singular fibers $\Omega_{X_y}$ is not the "right" thing to look at, so in some sense it is actually not a big deal that that's not what you get in the function.

One can actually look at the objects that come from the Deligne-Du Bois complex and do Hodge theory for singular varieties. Then the restriction becomes a little tricky, because those are objects in a derived category and restriction is not exact, so you need to do something else. Completion along the fiber gives the right thing, but I don't know if there is a semi-continuity theorem using completion instead of restriction. That might be an interesting question to contemplate.