How do we know that the intrinsic brightness of Cepheid variables corresponds to their period? As I understand it, beyond parallax the next way of estimating the distance to astronomical objects is by using a relationship between the period and intrinsic luminosity of Cepheid variables.  How do we know that relationship?  Are there some close enough that we used parallax to validate it, or is there a theoretical reason, or something else?
This relationship was discovered empirically by Henrietta Leavitt by comparing the apparent magnitudes of stars in the Magellanic clouds. Since these clouds are far away (200,000 ly) and relatively small (7,000 ly), the difference in distances to the 47 different Cephid variables she observed could account for only a small difference in brightness.  By plotting the brightness against the period she noticed that there was a relationship, cephids with longer periods were brighter.

As you guessed, modern measurements are made via parallax for close stars. The Hipparcos satellite compiled a vast list of parallaxes, and by measuring a large number of periods, we can calculate the relation.