We present a new and elegant cycle vector space algorithm that runs inO(n2.α)steps and needsO:lpar;nn)space for enumerating all simple cycles of a planar graph withnvertices, whereαis the total number of simple cycles in the graph Unlike backtrack algorithms, cycle vector space algorithms for this problem are suitable for parallelization. A parallel version of this algorithm alone with a parallel version of Syslo'sO(n.α)step algorithm for the same problem are on an exclusive-read, exclusive-write parallel RAM model withpprocessors. The results of an implementation of our parallel algorithm on a meshconnected SIMD computer are also presented.