Sulfide perovskites (ABX3) are under increasing investigation for use in photovoltaic, optoelectronic, dielectric, and thermoelectric devices due to their favorable band gaps, dynamical properties, environmental stability, and structural diversity. A key material parameter to optimize in such devices is the constituent materials' coefficient of thermal expansion (CTE) in order to minimize thermomechanical stress during fabrication and operation. This can be done by avoiding materials that have large CTE mismatch or by compensating positive thermal expansion by including materials with negative thermal expansion (NTE). Here, we evaluate the CTE of (edge-connected) α- and (corner-connected) β-SrZrS3 with density functional theory and the self-consistent quasiharmonic approximation. We find that both materials exhibit positive thermal expansion at 0 GPa and host pressure-induced negative thermal expansion. The β phase has a smaller CTE (37 × 10-6 K-1) at room temperature and ambient pressure and also has a larger NTE response under pressure due to its more flexible corner-connected framework structure. We use our findings to suggest that corner-shared motifs should be prioritized over edge- or face-shared octahedral networks to maximize NTE arising from vibrational (phononic) mechanisms.