  In this paper, we study the distribution of parallelograms and rhombi in a
given set in the plane over arbitrary finite fields $\mathbb{F}_q^2$. As an
application, we improve a recent result due to Fitzpatrick, Iosevich, McDonald,
and Wyman (2021) on the Vapnik-Chervonenkis dimension of the induced distance
graph. Our proofs are based on the discrete Fourier analysis.
