Since, tangent at $P$ and $Q$ on the parabola meet in $T$

Taking parametric co-ordinates of the point of tangency, let the coordinates of $P$ and $Q$ are $\left(a{t}_1^2, 2a{t}_1\right)$ and $\left(a{t}_2^2, 2a{t}_2\right)$ respectively, then coordinates of $T$ are $\left\{a{t}_1{t}_2, a\left(t_1+t_2\right)\right\}$

Now $S$ being the focus of the standard parabola $y^2=4ax$ i.e. co-ordinates of $S$ being $\left(a,0\right)$.

By distance formula, 

$\therefore SP=\sqrt{{\left(a-a{t}_1^2\right)}^2+{\left(0-2a{t}_1^2\right)}^2}=\sqrt{a^2+a^2{t}_1^4-2a^2{t}_1^2+4a^2{t}_1^2}=\sqrt{a^2+a^2{t}_1^4+2a^2{t}_1^2}=a\left(1+t_1^2\right)$

Similarly, $SQ=\sqrt{{\left(a-a{t}_2^2\right)}^2+{\left(0-2a{t}_2^2\right)}^2}=\sqrt{a^2+a^2{t}_2^4-2a^2{t}_2^2+4a^2{t}_2^2}=\sqrt{a^2+a^2{t}_2^4+2a^2{t}_2^2}=a\left(1+t_2^2\right)$

and $ST=\sqrt{a^2{\left(1-t_1{t}_2\right)}^2+a^2{\left(t_1+t_2\right)}^2}$
$=\sqrt{a^2\left(1+t_1^2+t_2^2+t_1^2{t}_2^2\right)}$
$=a\left(1+t_1^2\right)a\left(1+t_2^2\right)=SP\times SQ$
Thus, $SP, ST, SQ$ are in GP