Parallel lines containing a subset with even cardinality

Question

For each $\alpha \in \mathbf{R}\cup \{\infty\}$, let $\mathscr{L}_\alpha$ denote the collection of lines $\ell$ of $\mathbf{R}^2$ with slope $\alpha$. More explicitly: if $\alpha \in \mathbf{R}$, then $\ell \in \mathscr{L}_\alpha$ if and only if $\ell=\{(x,y) \in \mathbf{R}^2: y=\alpha x+\beta\}$, for some $\beta \in \mathbf{R}$; and $\ell\in \mathscr{L}_\infty$ if and only if $\ell=\{(x,y) \in \mathbf{R}^2: x=\gamma\}$, for some $\gamma \in \mathbf{R}$.

Now, let us fix a finite subset $\mathscr{P}:=\{P_1,\dotsc,P_k\}\subseteq \mathbf{R}^2$, where $k$ is a given positive even integer. For each $\alpha \in \mathbf{R}\cup \{\infty\}$, let $N(\alpha, \mathscr{P})$ be the number of points contained in lines $\ell$ with slope $\alpha$ which contain at least two points of $\mathscr{P}$, that is, $$ N(\alpha, \mathscr{P}):=|\mathscr{P} \cap L(\alpha,\mathscr{P})|, \quad \text{ where }\quad L(\alpha,\mathscr{P}):=\{\ell\in \mathscr{L}_\alpha: |\,\ell \cap \mathscr{P}\,|\ge 2\}. $$

Question. Is it true that there exists $\alpha \in \mathbf{R}\cup \{\infty\}$ such that $N(\alpha,\mathscr{P})$ is an even positive integer?

Some observations: if the points in $\mathscr{P}$ are collinear, the answer is clearly affirmative. Otherwise, it is also affirmative if $\mathscr{P}\subseteq \{a,b\}\times \mathbf{R}$ for some distinct $a,b$: indeed, in such case, there would be a maximal slope $\alpha \in \mathbf{R}$ connecting two points with different abscissas so that $N(\alpha,\mathscr{P})=2$. Moreover, it may be possible that $|L(\alpha,\mathscr{P})| \neq 1$ for every $\alpha$: for, let $\mathscr{P}$ be the grid $\{1,\dotsc,2n\}^2$, for some $n\ge 2$. Differently from the estimates on point-line incidences, here we are interested in the parity of the latter ones.

Answer 1

It was conjectured by Dumitrescu and Toth in "Distinct triangle areas in a planar point set" (see Problem 1 at the end) that for every large enough $k$ for any $\mathscr P$ there is an ordinary direction $\alpha$ such that for all $\ell\in L(\alpha,\mathscr P)$ we have $|\ell\cap \mathscr P|=2$.
This would imply an affirmative answer to your question, as in this case $N(\alpha,\mathscr P)=2|L(\alpha,\mathscr P)|$.
Though there is no implication in the reverse direction, I don't see how one could exploit parity, so I think that your problem is just as hard as theirs.