There are only three 6-SPS parallel manipulators with triangular base and platform, i.e., the octahedral, the flagged, and the partially flagged, which are studied in this paper. The forward kinematics of the octahedral manipulator is algebraically intricate, while those of the other two can be solved by three trilaterations. As an additional nice feature, the flagged manipulator is the only parallel platform for which a cell decomposition of its singularity locus has been derived. Here, we prove that the partially flagged manipulator also admits a well-behaved decomposition, technically called a stratification, some of whose strata are not topological cells, however. Remarkably, the adjacency diagram of the 5-D and 6-D strata (which shows what 5-D strata are contained in the closure of a 6-D one) is the same as for the flagged manipulator. The availability of such a decomposition permits devising a redundant 7-SPS manipulator, combining two partially flagged ones, which admits a control strategy that completely avoids singularities. Simulation results support these claims.