First of all, your picture looks excellent - nice work.
Now, to switch the order of integration, we will be integrating in the $yz$ plane first then integrating over $x$. So, we need to think about what $yz$ slices of our region look like. Key observation: they are triangles:
I've labelled all the key points of interest; we first fix $x$, which gives us a triangular cross section. Then, we fix $z$ and vary $y$ (the horizontal line). Its first point of contact with the solid is at $y=x^2$. It leaves the solid when $y=4-4z$ (i.e. when it hits the surface $z=1-y/4$). These are your $y$ bounds. Finally we must vary $z$ within the triangle - it starts at $0$, and ends at the "top" vertex of the triangle. There, $y=x^2$, so we have $z=1-x^2/4$. Thus we have:
The general strategy is to fix the last variable, then work over the cross section formed by the other 2 variables. Hope this helps!