  False theta functions form a family of functions with intriguing modular
properties and connections to mock modular forms. In this paper, we take the
first step towards investigating modular transformations of higher rank false
theta functions, following the example of higher depth mock modular forms. In
particular, we prove that under quite general conditions, a rank two false
theta function is determined in terms of iterated, holomorphic, Eichler-type
integrals. This provides a new method for examining their modular properties
and we apply it in a variety of situations where rank two false theta functions
arise. We first consider generic parafermion characters of vertex algebras of
type $A_2$ and $B_2$. This requires a fairly non-trivial analysis of Fourier
coefficients of meromorphic Jacobi forms of negative index, which is of
independent interest. Then we discuss modularity of rank two false theta
functions coming from superconformal Schur indices. Lastly, we analyze
$\hat{Z}$-invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing ${\tt
H}$-graphs. Along the way, our method clarifies previous results on depth two
quantum modularity.
