Modularity of a certain continued fraction of Ramanujan

We apply the methods of Lee and Park to study a certain continued fraction H(tau) of Ramanujan, which is a particular case of his identity written in one of his notebooks. We prove that H(tau) can be expressed in terms of an eta-quotient s(tau), which is a generator for the field of all modular functions on the congruence subgroup Gamma(0)(12). We also show that there is a modular equation for s(tau) and H(tau) of level n for all positive integers n, and provide explicitly the modular equations of level p for primes p <= 13.