The solution of uniform bicubic B-spline curve/surface fitting problem is considered. Based on the matrix perturbation method, this paper first presents a novel approximateO(n/p)-time parallel B-spline curve fitting algorithm for finding the correspondingncontrol points that interpolate thosendata points on a linear array processor withpprocessors, wherep≤n. Givenm×ndata points, we then present anO(mn/(p1p2))-time parallel algorithm for solving the uniform bicubic B-spline surface fitting problem on ap1×p2mesh-connected computer, wherep1≤mandp2≤n. The relative error analyses of our two stable and cost-optimal parallel solvers are also given. When settingp1=mandp2=n, a constant-time parallel solver for B-spline surface fitting can be derived; this time- and cost-optimal result is a direct method, in contrast to the parallel iterative method of Chenget al.(Parallel B-spline surface interpolation on a mesh-connected processor array,J. Parallel Distrib. Comput.24, 2 (1995), 224–229).