Remote center-of-motion (RCM) parallel manipulators (PMs) are fit for robotized minimally invasive surgery (MIS). RCM PMs with fixed linear actuators have the advantages of high stiffness, reduced moving mass, and higher rigidity and load capacity. However, there are very few available architectures of these types of PMs. Using the Lie group algebraic properties of the set of rigid-body displacements, this paper proposes a new family of RCM PMs with fixed linear actuators for MIS. The general motion with a remote center has four degrees-of-freedom (DOF) and is produced by the in-series concatenation of a spherical S pair and a prismatic P pair and, therefore, is said to be SP equivalent. The SP-equivalent PMs can be used in minimally invasive surgery. First, the kinematic bonds of limb chains and their mechanical generators for SP-equivalent RCM PMs are presented. Limb chains with fixed linear actuators are then derived using the closure of products in subgroups. Structural conditions for constructing an SP-equivalent RCM PM with linear fixed actuators are revealed. Helical pairs are introduced to remove a local rotation and yield a 360-deg-rotation capability of the moving platform. Numerous new architectures with practical potential are presented.