Cummins and Pauli have calculated generators for the function fields of all congruence subgroups of $\text{PSL}_2(\mathbb{F}_p)$ of genus $\le 24$ in:

http://www.mathstat.concordia.ca/faculty/cummins/congruence/

I haven't looked at this for a few months but I believe that the companion paper http://www.emis.de/journals/EM/expmath/volumes/12/12.2/pp243_255.pdf discusses the generators.  In the meantime there is a paper by Yifan Yang "Defining equations of modular curves" Advances in Mathematics
Volume 204, Issue 2, 20 August 2006, Pages 481-508

which gives tables of equations for many modular curves, and discusses a methodology for finding "good" equations (i.e. those with small coefficients and a small number of terms in the defining polynomials)