In this paper we study the homotopy theory of parameterized spectrum objects in the $\infty$-category of $(\infty, 2)$-categories, as well as the Quillen cohomology of an $(\infty, 2)$-category with coefficients in such a parameterized spectrum. More precisely, we construct an analogue of the twisted arrow category for an $(\infty,2)$-category $\mathbb{C}$, which we call its twisted 2-cell $\infty$-category. We then establish an equivalence between parameterized spectrum objects over $\mathbb{C}$, and diagrams of spectra indexed by the twisted 2-cell $\infty$-category of $\mathbb{C}$. Under this equivalence, the Quillen cohomology of $\mathbb{C}$ with values in such a diagram of spectra is identified with the two-fold suspension of its inverse limit spectrum. As an application, we provide an alternative, obstruction-theoretic proof of the fact that adjunctions between $(\infty,1)$-categories are uniquely determined at the level of the homotopy $(3, 2)$-category of $\mathrm{Cat}_{\infty}$.