Low compression multi-piece golf balls having a solid core comprising at least one layer made of a plasticized thermoplastic composition are provided. Preferably, the ball has a compression of less than 60 and more preferably less than 50. The ball includes a cover in addition to the core sub-assembly. For example, two-piece and three-piece golf balls may be made. The plasticized thermoplastic composition preferably comprises: a) ethylene acid copolymer, b) plasticizer, and c) cation source. Preferably, a fatty acid ester such as butyl oleate is used as the plasticizer. In two-layered core constructions, the outer core layer is preferably formed from a thermoset composition such as polybutadiene rubber. The core layers may have different hardness levels. The core structure and resulting ball have relatively good resiliency at given compressions.