  We study algebraic relations among postcritically finite (PCF) parameters in
the family $f_c(z) = z^2 + c$. Ghioca, Krieger, Nguyen and Ye proved that an
algebraic curve in $\mathbb{C}^2$ contains infinitely many PCF pairs $(c_1,
c_2)$ if and only if the curve is special (i.e., the curve is a vertical or
horizontal line through a PCF parameter, or the curve is the diagonal). Here we
extend this result to subvarieties of $\mathbb{C}^n$ for any $n\geq 2$.
Consequently, we obtain uniform bounds on the number of PCF pairs on
non-special curves in $\mathbb{C}^2$ and the number of PCF parameters in real
algebraic curves in $\mathbb{C}$, depending only on the degree of the curve. We
also compute the optimal bound for the general curve of degree $d$.
