$P$ and $Q$ are any two points on the parabola where $TP$ is the tangent and $TQ$ is parallel to the axis of symmetry. Prove that the locus of the midpoint of $TP$ is the directrix of the parabola.
Let $P = (ap^2, 2ap), Q = (aq^2, 2aq)$, and the equation of the parabola be $y^2 = 4ax$, then we get the equation of $TP: y = \frac{x}{p} + ap \tag1$ and $TQ: y = 2aq \tag2$
Solving $(1)$ and $(2)$ gives $x = 2apq - ap^2, y = 2aq$.
Then the midpoint of $TP = (apq, ap + aq)$ and basically I'm stuck here.
I think some information is missing in your question. The result at which you have arrived is correct but proving that the locus is the directrix of the paraola requires the points $P$ and $Q$ to lie on the focal chord of the parabola.Since the points , which lie on focal chord of the parabola satisfy $pq=-1$, the locus turns out to be $x=-a$ which is the directrix.
The claim that $pq=-1$ can be proven very easily, so I am leaving it.