Persistent spin textures are highly desirable for applications in spintronics, as they allow for long carrier spin lifetimes. However, they are also rare, as they require a delicate balance between spin-momentum coupling parameters. We used density functional theory simulations to predict the possibility of achieving these desirable spin textures through the application of uniaxial stress. Hybrid organic-inorganic perovskite MPSnBr3 (MP = CH3PH3) is a ferroelectric semiconductor which exhibits persistent spin textures near its conduction band minimum and mostly Rashba type spin textures in the vicinity of its valence band maximum. Application of uniaxial stress leads to the gradual evolution of the valence band spin textures from mostly Rashba type to quasipersistent ones under a tensile load and to pure Rashba or quasipersistent ones under a compressive load. The material exhibits flexibility, a rubber-like response, and both positive and negative piezoelectric constants. A combination of such properties may create opportunities for flexible and rubbery spintronic devices.