We show how to transform the B-spline curve and surface fitting problems into suffix computations of continued fractions. Then a parallel substitution scheme is introduced to compute the suffix values on a newly proposed mesh-of-unshuffle network. The derived parallel algorithm allows the curve interpolation through n points to be solved in Ο(log n ) time using Θ n /log n ) processors and allows the surface interpolation through m x n points to be solved in Ο(log m log n ) time using Θ ( mn /(log m log n )) processors. Both interpolation algorithms are cost-optimal for their respective problems. Besides, the surface fitting problem can be even faster solved in Ο(log m + log n ) time if Θ( mn ) processors are used in the network.