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Using stellar population synthesis models to infer star formation histories (SFHs), we analyse photometry and spectroscopy of a large sample of quiescent galaxies which are members of Sunyaev-Zel'dovich (SZ)-selected galaxy clusters across a wide range of redshifts. We calculate stellar masses and mass-weighted ages for 837 quiescent cluster members at 0.3 < z < 1.4 using rest-frame optical spectra and the Python-based Prospector framework, from 61 clusters in the SPT-GMOS Spectroscopic Survey (0.3 < z < 0.9) and 3 clusters in the SPT Hi-z cluster sample (1.25 < z < 1.4). We analyse spectra of subpopulations divided into bins of redshift, stellar mass, cluster mass, and velocity-radius phase-space location, as well as by creating composite spectra of quiescent member galaxies. We find that quiescent galaxies in our dataset sample a diversity of SFHs, with a median formation redshift (corresponding to the lookback time from the redshift of observation to when a galaxy forms 50% of its mass, t$_{50}$) of $z=2.8\pm0.5$, which is similar to or marginally higher than that of massive quiescent field and cluster galaxy studies. We also report median age-stellar mass relations for the full sample (age of the Universe at $t_{50}$ (Gyr) = $2.52 (\pm0.04) - 1.66 (\pm0.11)$ log$_{10}(M/10^{11} M\odot))$ and recover downsizing trends across stellar mass; we find that massive galaxies in our cluster sample form on aggregate $\sim0.75$ Gyr earlier than lower mass galaxies. We also find marginally steeper age-mass relations at high redshifts, and report a bigger difference in formation redshifts across stellar mass for fixed environment, relative to formation redshifts across environment for fixed stellar mass., 35 pages, 23 figures, 5 tables. Submitted to The Astrophysical Journal. Comments and feedback welcome