Intuitively, I was thinking we can approach this the same way as two lines in 3D space, where they may not intersect nor be parallel.
I think I've come up with the example of ${(0,x,y,0):x,y \in R},{(1,x,0,y):x,y \in R}$, where the planes are not parallel and also don't intersect. However, I'm not sure how to develop this into a proof that it's true in general.
A proof of this statement comes directly form Rouché-Capelli theorem; two planes are described each by two equations, when you intersect them you find a 4x4 matrix A associated to the sistem AX=b. You can suppose that only three out the four equations are linearly independent, so you can choose b linearly independent from the columns of A , and the sistem has no solution, so the planes don’t intersect. Now there is no containment with the space of directions of the two plane but then intersection in not the zero space. So they aren’t parallel.