This site https://www.ics.uci.edu/~eppstein/junkyard/parallel-postulate.html lists from the book "The Foundations of Geometry and the Non-Euclidean Plane" by George E. Martin 26 propositions which are suppose to be equivalent to the parallel postulate. In particular Proposition I states:
There exists a circle passing through any three noncollinear points.
However, I would say this is valid in spherical geometry, so it can't be equivalent to the parallel postulate since the parallel postulate is not valid in spherical geometry.
As the linked site says, these are equivalents to the parallel postulate within absolute geometry. Spherical geometry is not a model of absolute geometry, so this equivalence does not apply in that context.
(Absolute geometry is roughly "Euclidean geometry without assuming the parallel postulate", but the parallel postulate is not the only part of Euclidean geometry that fails in spherical geometry. There are different ways to precisely define "absolute geometry" which are not all exactly equivalent (and I'm not sure which one Martin's book uses), but roughly it means you are in either Euclidean geometry or hyperbolic geometry.)