The ansatz $a \equiv c \equiv 1 \mod 2021$ and $2021 \mid b$ leads to the equation $a^2+b^2=c^2$. The solutions have the well-known classification as $a=u^2-v^2, b=2uv, c=u^2+v^2$ with $u,v$ coprime. We only need to choose $u>v$ coprime with $u \equiv 1 \mod 2021$ and $2021 \mid v$. It is clear that this can be done in infinitely many ways.