Why does the slope of parent-offspring regression equals the heritability in the narrow sense?
Well, I think I found the very simple mistake I made…

Looking again in my equations, I realize that (for some reason) $cor = 2 \cdot \frac{\sigma_A^2}{\sigma}$

And looking at this website, I see that the slope of the parent-offspring regression is $\frac{h_N^2}{2} = slope$

Here was my mistake!