  A probabilistic frame is a Borel probability measure with finite second
moment whose support spans $\mathbb{R}^d$. A Parseval probabilistic frame is
one for which the associated matrix of the second moments is the identity
matrix in $\mathbb{R}^d$. Each probabilistic frame is canonically associated to
a Parseval probabilistic frame. In this paper, we show that this canonical
Parseval probabilistic frame is the closest Parseval probabilistic frame to a
given probabilistic frame in the $2-$Wasserstein distance. Our proof is based
on two main ingredients. On the one hand, we show that a probabilistic frame
can be approximated in the $2-$Wasserstein metric with (compactly supported)
finite frames whose bounds can be controlled. On the other hand, we establish
some fine continuity properties of the function that maps a probabilistic frame
to its canonical Parseval probabilistic frame. Our results generalize similar
ones for finite frames and their associated Parseval frames.
