How do I make the gates in my fence work properly?

To mount the hinges parallel to the slope, stick that post in perpendicular to the slope.

Or you could have a sliding gate that goes sideways instead of swinging

Or, more complicated, a parallelogram that lifts one end as it swings kind of like this picture.

Or simply swing the gate out over the slope instead of back into it.

I searched for “hinges for gates on slopes” to get this and this … and those were just in the top 5 results. I’ll bet you can find some other good solutions.

Of course, you’ll need to keep in mind that any offset that causes the gate to rise to clear the upward slope is also going to get gravity involved: the more that the gate opens inward, the heavier it gets. So if it’s an iron or heavy wood gate to begin with, that’s a definite consideration.

This site offers a number of attractive solutions for gates on a slope.

Mount the gate vertically with a counterweight, so that it works like a garage door.

Rolling gate like this

Mount the gate via the hinges to the downhill post.

@kritiper it is a pair so that is really not an option.

One thing I did notice is that the vast majority, if not all, the offset hinge options are in the UK.

Run that section of the fence where the gate will be along the width of the slope instead of against it.