<p>This paper is devoted to time-space finite element analysis for time-space fractional parabolic problems with spectral fractional Laplacian under weak regularity assumptions. First, a discrete spectral fractional Laplacian is introduced based on matrix transfer technique on quasi-uniform meshes, and time-space finite element discrete scheme for time-space fractional parabolic problem is established using continuous time Petrov-Galerkin method and spatially conforming finite element method. In order to capture the weak singular behavior of the exact solution and improve the global convergence order of the discrete scheme, time-stepping finite element scheme is implemented on graded meshes. Afterwards, the well-posedness of the approximate solutions is discussed based on the continuity and coercivity given by Caputo derivative. In addition, the convergence analysis of the fully discrete scheme under weak regularity conditions is investigated by means of time-space projection estimates and consistency error estimate. Finally, a fast time-stepping finite element scheme is designed and numerical examples are provided to validate our theoretical results. Numerical experiments show that the error order is better than the present theoretical analysis and the use of graded meshes does improve the global numerical accuracy.</p>