When a ray passes through focus does it move parallel to the principal axis after it reflects in both the cases if yes then why?

Consider an incoming ray which is parallel to the principal axis which hits a concave mirror at $X$.

The normal to this mirror at $X$ passes through the centre of curvature of the mirror $C$.

$\frac {h}{CP'}= \tan \alpha, \, \frac {h}{FP'}= \tan 2\alpha, \, $

For small $\alpha$ ie incoming ray close to the principal axis

$CP' \approx CP,\, FP' \approx FP,\, \tan \alpha \approx \alpha,\, \tan 2\alpha \approx 2\alpha$ where $P$ is the pole of the mirror.

$\Rightarrow CP \approx 2 FP$

$CP$ is a property of the mirror and $FP$, the focal length of the mirror, is thus (approximately) independent of the angle $\alpha$ as long as $\alpha$ is small.

If $\alpha$ is not small then a mirror defect called spherical aberration occurs as shown below.

(On Desmos)

You often see this whilst having a drink with the bright line being called a caustic curve.