A boundary element collocation method for some parabolic pseudodifferential equations is examined. The two-dimensional heat conduction problem with vanishing initial condition and a given Neumann or Dirichlet type boundary condition is chosen as the basic model. The parabolic pseudodifferential operators are contained in the boundary integral equations of the first kind (being the single layer and the hypersingular heat operator equations). The collocation schema is reduced to an equivalent Galerkin type problem. For the approximation of the solution tensor products of spline functions are used. The spatial variable is approximated by periodic smoothest splines of arbitrary high odd degree and the time variable is approximated by piecewise linear continuous splines. Stability and convergence of the method are proved with respect to some anisotropic Sobolev norms. Some results of this paper are more general than those obtained by \textit{M. Costabel} and \textit{J. Saranen} [Numer. Math. 84, No.~3, 417-449 (2000; Zbl 0967.65103)].