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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?

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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 and relatively small, the differences between the stars' distances to us are small compared to the distance from us to the cloud.

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.

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