The following image is from this answer for the question How to prove the parametric equation of an ellipse?, and shows an ellipse (red) and two concentric circles (black) with diameters equal to the major and minor axes of the ellipse, with centre at the centre of the ellipse:
The coordinates of the point $P$ in the parametric form is $(a\cos\theta,b\sin\theta)$ where $\theta$ is the angle $AOH$ and $a,b$ are the lengths of the semi-major and semi-minor axes respectively.
I can understand, since $AH$ is perpendicular to $DE$, the $x$ coordinate of $A$ is same as that of $P$ which is $a\cos\theta$.
But, I am unable to understand why the $y$ coordinate of the point $P$ is $b\sin\theta$. According to the previously mentioned question/answer, I think we are supposed to understand like this - since $BP$ is parallel to $DE$ the $y$ coordinates of $B$ and $P$ are the same, which is $b\sin\theta$. But, I am unable to understand this fact due to the following:
Why must $BP$ be parallel to $DE$?
A line perpendicular to $AH$ passing through $P$ (or a line parallel to $DE$ passing through $P$) will pass for sure through the inner circle. But why should this point lie on the line $AO$?
To be concise, kindly explain why is the $y$ coordinate of a point in a ellipse in the parametric form $b\sin\theta$?
Thank you in advance.
The question you mentioned defined the ellipse as the locus of point $P$ and asked to prove that this was in agreement with the two-foci definition of ellipse. That seems to have little to do with your actual question.
If you define $P$ as the point with coordinates $(a\cos\theta,b\sin\theta)$ then it is obvious that $PB$ is parallel to the $x$-axis, because the coordinates of $B$ are $(b\cos\theta,b\sin\theta)$: points $P$ and $B$ have the same $y$-coordinate.