On the number of integer points with even coordinates in the interior of a lattice triangle

Question

I am looking at triangles with vertices $(0,1)$, $(0,2k+1)$, $(2l+1, 2k+1)$ where $k$ and $l$ are integers and $2l$ and $2k+1$ are prime to each other. I am trying to describe $P\pmod2$, where $P$ the number of integer points $(x,y)$ with $x\equiv 0 \pmod2$ and $y \equiv 0 \pmod2$ lying in the interior of such triangles.
For $k=1$ or $k=2$, it is easy to work out a formula for $P$, and in general it looks like the behaviour of $P$ depends only on $l$ and $k \pmod{2l}$. Do you know any result or trick that could help me with this problem ?