Maximizing a non-negative, monontone, submodular function $f$ over $n$ elements under a cardinality constraint $k$ (SMCC) is a well-studied NP-hard problem. It has important applications in, e.g., machine learning and influence maximization. Though the theoretical problem admits polynomial-time approximation algorithms, solving it in practice often involves frequently querying submodular functions that are expensive to compute. This has motivated significant research into designing parallel approximation algorithms in the adaptive complexity model; adaptive complexity (adaptivity) measures the number of sequential rounds of $\text{poly}(n)$ function queries an algorithm requires. The state-of-the-art algorithms can achieve $(1-\frac{1}{e}-\varepsilon)$-approximate solutions with $O(\frac{1}{\varepsilon^2}\log n)$ adaptivity, which approaches the known adaptivity lower-bounds. However, the $O(\frac{1}{\varepsilon^2} \log n)$ adaptivity only applies to maximizing worst-case functions that are unlikely to appear in practice. Thus, in this paper, we consider the special class of $p$-superseparable submodular functions, which places a reasonable constraint on $f$, based on the parameter $p$, and is more amenable to maximization, while also having real-world applicability. Our main contribution is the algorithm LS+GS, a finer-grained version of the existing LS+PGB algorithm, designed for instances of SMCC when $f$ is $p$-superseparable; it achieves an expected $(1-\frac{1}{e}-\varepsilon)$-approximate solution with $O(\frac{1}{\varepsilon^2}\log(p k))$ adaptivity independent of $n$. Additionally, unrelated to $p$-superseparability, our LS+GS algorithm uses only $O(\frac{n}{\varepsilon} + \frac{\log n}{\varepsilon^2})$ oracle queries, which has an improved dependence on $\varepsilon^{-1}$ over the state-of-the-art LS+PGB; this is achieved through the design of a novel thresholding subroutine.