Take a unit tangent p-vector $a$ based at $y \in Y$ (you need the volume form on $Y$ to say what a unit p-vector is). At each point $x$ of the fiber $f^{-1}(y)$ consider the contraction of your volume form on $X$ with any p-vector that projects down to $a$ by the differential $Df$. Note that the pullback of this contracted form to the fiber (you only evaluate it on $n-p$ vectors tangent to the fiber) does not depend on the choice of the p-vectors that project to $a$.

This is basically the pushforward construction for forms, which also works for densities.

Take a unit tangent p-vector $a$ based at $y \in Y$ (you need the volume form on $Y$ to say what a unit p-vector is). At each point $x$ of the fiber $f^{-1}(y)$ consider the contraction of your volume form on $X$ with any p-vector $b$ in $\Lambda^p(T_xX)$ that projects down to $a$ by the linear map $D_xf$. Note that the pullback of this contracted form to the fiber (you only evaluate it on $n-p$ vectors tangent to the fiber) does not depend on the choice of the p-vectors that project to $a$.

This is basically the pushforward construction for forms, which also works for densities.