The general equation of parabola is ${(y-h)}^2=4a(x-k)$ where $\left(h,k\right)$ is the vertex.

Given vertex $\left(1, -2\right)$ and focus $\left(\frac{5}{4}, -2\right)$. Hence, $a=\frac{5}{4}-1=\frac{1}{4}$

Substituting the value in equation,

$\Rightarrow {(y+2)}^2=4\times \frac{1}{4}(x-1)$

  $\Rightarrow {(y+2)}^2=(x-1)$

$\therefore \left(10, 1\right)$ satisfies the equation.