Given,

Equation of parabola is $y^2=16x$, so on comparing with $y^2=4ax$ we get $a=4$,

Now given $P=\left(4,8\right)$ and $R=\left(16,16\right)$,

Now we know that parametric point of given parabola will be $\left(a{t}^2,2at\right)\equiv \left(4t^2,8t\right)$

Now parameter for $\left(4,8\right)$ on comparing will be $t=1$ and similarly for point $\left(16,16\right)$ $t=2$

Now we know that for any focal chord the relation in parameter is $t_1{t}_2=-1$

So point $Q$ will be $\left(4,-8\right)$ and similarly point $T$ will be $\left(1,-4\right)$ using the relation $t_1{t}_2=-1$,

Now using distance formula we get $QT=5$