  This work systematically investigates the phase diagram of a parallel
double-quantum-dot Andreev molecule, where the two quantum dots are coupled to
a common superconducting lead. Using the numerical renormalization group
method, we map out the evolution of the ground state across a wide parameter
space of level detunings, size of the superconducting gap, lead couplings, and
inter-dot coupling strength. The intricate phase diagrams feature singlet,
doublet, and a relatively uncommon triplet ground states, with the latter being
a distinct signature of strong lead-mediated interactions between the quantum
dots. We benchmark the applicability of simplified effective models, including
the atomic limit and zero-bandwidth approximations, in capturing the complex
behavior of this parallel configuration. Our analysis reveals severe
limitations of these models, underscoring the necessity for maximal caution
when extrapolating beyond their tested validity. In particular, all effective
models except for the extended version of the zero-bandwidth approximation
failed in reproducing the triplet ground state and made several false
predictions. These findings provide crucial insights for interpreting
experimental observations and designing superconducting devices based on
quantum-dot architectures.
