Brolin's theorem states that for a monic polynomial f on the complex plane of degree d greater than or equal to 2, for a non-exceptional point a, the backwards orbit of a equidistributes on the Julia set of f [1]. Tortrat [9] proved a version of Brolin's theorem for periodic points. Drasin and Okuyama [3] proved a rate of convergence result for Brolin's theorem, and we use some of their work to prove a similar result for the periodic version of Brolin's theorem whenever f is a quadratic polynomial with parameter c in the main cardioid of the Mandelbrot set.