I am guessing that the slant height of an oblique prism is defined as the length of each segment connecting corresponding points of the two bases rather than the distance up the middle of any of the lateral faces, since all 3 pairs of lateral faces can be different in a parallelepiped. Is this correct?
I realized that an oblique prism can either slant in only one direction (so its faces include 2 parallelograms and 4 rectangles, as in the picture) or slant in both directions (a parallelepiped with 6 parallelograms as faces). Is there a formula for the surface area that always applies in the first case? I'm guessing that for the surface area of a parallelepiped one has to calculate the area of each parallelogram separately and add them together since the 6 parallelograms are not guaranteed to all be congruent.
I believe the lengths of the segments connecting the corresponding points would be the "lateral edges." I don't think the term "slant height" could apply here since the sides are not triangles, but the dimension that would correspond to slant height of a pyramid would, I believe, be the height of each of the parallelogram faces (which, as you say, can be different from each other in an oblique prism).