The existing remote center of motion (RCM) parallel mechanisms (PMs) can only have single-mode operation. To provide more modes, this paper designed a new family of RCM PMs with 3R1T，2R1T，and 1R1T modes. First, this paper proposed a class of side chains with a movable rotational center, and integrated it into the parallel mechanism as a branch of the parallel mechanism. Then, based on the screw theory, verified the influence of the branch rotation center of gravity in parallel mechanisms when determining degrees of freedom. Second, the theory of multi-mode parallel mechanisms is proposed. Based on this theory, for RCM parallel mechanisms with DOFs—3R1T, 2R1T, and 1R1T—separate targeted analyses are conducted. Meanwhile, the feasible branch chain configurations and essential assembly requirements for multi-mode RCM mechanisms are systematically and comprehensively deduced. Finally, by integrating a branch chain with movable rotational center properties, a multi-mode RCM parallel mechanism capable of achieving 3R1T, 2R1T, and 1R1T configurations is proposed. On this basis, targeted selection of multi-mode configurations is conducted for the proposed mechanism, and verification tests on its motion modes are carried out—results of which confirm the rationality and correctness of the multi-mode design concept. Notably, the developed multi-degree-of-freedom RCM parallel mechanism allows for switching between different degrees of freedom without the need for disassembly and reassembly of components, realizing continuous adjustment of motion performance while ensuring structural integrity and operational efficiency.