The zero-temperature phase diagram of the spin-$frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{perp}$ model on an $AA$-stacked square-lattice bilayer is studied using the coupled cluster method implemented to very high orders. Both nearest-neighbor (NN) and frustrating next-nearest-neighbor Heisenberg exchange interactions, of strengths $J_{1}>0$ and $J_{2} equiv kappa J_{1}>0$, respectively, are included in each layer. The two layers are coupled via a NN interlayer Heisenberg exchange interaction with a strength $J_{1}^{perp} equiv delta J_{1}$. The magnetic order parameter $M$ (viz., the sublattice magnetization) is calculated directly in the thermodynamic (infinite-lattice) limit for the two cases when both layers have antiferromagnetic ordering of either the N{e}el or the striped kind, and with the layers coupled so that NN spins between them are either parallel (when $delta < 0$) or antiparallel (when $delta > 0$) to one another. Calculations are performed at $n$th order in a well-defined sequence of approximations, which exactly preserve both the Goldstone linked cluster theorem and the Hellmann-Feynman theorem, with $n leq 10$. The sole approximation made is to extrapolate such sequences of $n$th-order results for $M$ to the exact limit, $n to infty$. By thus locating the points where $M$ vanishes, we calculate the full phase boundaries of the two collinear AFM phases in the $kappa$--$delta$ half-plane with $kappa > 0$. In particular, we provide the accurate estimate, ($kappa approx 0.547,delta approx -0.45$), for the position of the quantum triple point (QTP) in the region $delta < 0$. We also show that there is no counterpart of such a QTP in the region $delta > 0$, where the two quasiclassical phase boundaries show instead an ``avoided crossing behavior, such that the entire region that contains the nonclassical paramagnetic phases is singly connected.
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