  We study $N$-point rational distance sets ($\textrm{RDS}(N)$) on the parabola
$y=x^2$. Previous approaches to the problem include efforts made using elliptic
curves and diophantine chains, with successful analysis for $N\leq 4$. We
extend the analysis for arbitrary $N$ by establishing a correspondence between
$\textrm{RDS}(N)$s and Pythagorean triplets. Our main result gives sufficient
and necessary conditions for the existence and nature of the $\textrm{RDS}(N)$s
for arbitrary $N$. Our approach also leads to an efficient computational
algorithm to construct new $\textrm{RDS}(N)$s, and we provide multiple new
examples of $\textrm{RDS}(N)$s for four and five points. The correspondence
with Pythagorean triplets also helps to study the density of the solutions and
we reproduce density results for $N=2$ and $3$.
