We develop a unified second-order parameterized complexity theory for spaces of integrable functions. This generalizes the well-established case of continuous functions. Specifically, we prove the mutual linear equivalence of three natural parameterizations of the space \(\textrm{L}^{p}\) of p-integrable complex functions on the real unit interval: (binary) \(\textrm{L}^{p}\) -modulus, rate of convergence of Fourier series, and rate of approximation by step functions.