When we refer to "parity" in Alcazar, we mainly refer to two types:
1. Exit parity. Every closed region must have an even number of used exits.
2. Checkerboard parity. Every time one moves, one must move from a light cell to a dark cell and vice versa.
There is a third type of parity in loop puzzles, which I call "flow parity", but this is not very easily applicable to Alcazar puzzles.
Exit parity shall be covered in a later post. For now, I shall focus only on checkerboard parity.
Take, for example, the following puzzle.
Now, say you've solved the puzzle to this point.
This logic doesn't just apply to the whole puzzle. It can also be applied to regions as well!
Thankfully, they cannot. If they did, then three would be dark and one would be light. However you draw paths in such a configuration, you will end up with one path that starts and ends on dark cells, and one that starts and ends on different-coloured cells. This would mean that you would use one more dark cell than light.
So you can only enter and exit it once. You can now draw in the light exit and solve the puzzle afterwards.
There is a general rule for any number of exits, though, and it goes as such (click to enlarge):