  Given $E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d$, with the finite
field $\mathbb{F}_q$ of order $q$ and the integer $d \ge 2$, we define the
two-parameter distance set as $\Delta_{d, d}(E)=\left\{\left(\|x_1-y_1\|,
\|x_2-y_2\|\right) : (x_1,x_2), (y_1,y_2) \in E \right\}$. Birklbauer and
Iosevich (2017) proved that if $|E| \gg q^{\frac{3d+1}{2}}$, then $ |\Delta_{d,
d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| \gg
q^{\frac{10}{3}}$, then $ |\Delta_{2, 2}(E)| \gg q^2$. In this paper, we
present extensions and improvements of these results.
