  In the recently discovered antiperovskite phosphide (Ca,Sr)Pd$_3$P,
centrosymmetric (CS) and non-centrosymmetric (NCS) superconducting phases
appear depending on the Sr concentration, and their transition temperatures
($T_\mathrm{c}$) differ by as much as one order of magnitude. In this study, we
investigated the superconducting properties and electronic band structures of
CS orthorhombic (CSo) (Ca$_{0.6}$Sr$_{0.4}$)Pd$_3$P ($T_\mathrm{c}$ = 3.5 K)
and NCS tetragonal (NCSt) (Ca$_{0.25}$Sr$_{0.75}$)Pd$_3$P ($T_\mathrm{c}$ =
0.32 K) samples with a focus on explaining their large $T_\mathrm{c}$
difference. Specific heat measurements indicated that CSo
(Ca$_{0.6}$Sr$_{0.4}$)Pd$_3$P was an s-wave superconductor in a
moderate-coupling regime with a 2$\Delta$$_0$/k$_B$$T_\mathrm{c}$ value of 4.0.
Low-lying phonons leading to the strong coupling in the structurally analogous
SrPt$_3$P were unlikely to be present in CSo (Ca$_{0.6}$Sr$_{0.4}$)Pd$_3$P.
Given that CSo (Ca$_{0.6}$Sr$_{0.4}$)Pd$_3$P and NCSt
(Ca$_{0.25}$Sr$_{0.75}$)Pd$_3$P exhibited similar Debye temperatures
($\Theta$$_D$) of approximately 200 K, the large $T_\mathrm{c}$ difference
could not be attributed to $\Theta$$_D$.$T_\mathrm{c}$ of each phase was
accurately reproduced based on the Bardeen-Cooper-Schrieffer (BCS) theory using
experimental data and the density of states of the Fermi level $N$(0)
calculated from their band structures. We concluded that the considerable
suppression of $T_\mathrm{c}$ in NCSt (Ca$_{0.25}$Sr$_{0.75}$)Pd$_3$P can be
primarily attributed to the decrease in $N$(0) associated with the structural
phase transition without considering the lack of inversion symmetry.
