We consider an initial boundary value problem on an interval for singularly perturbed parabolic PDEs with convection. The highest space derivative in the equation is multiplied by the perturbation parameter $\eps$, $\eps \in (0,1]$. Solutions of well-known classical numerical schemes for such problems do not converge $\eps$-uniformly (the errors of such schemes depend on the value of the parameter $\eps$ and are comparable with the solution itself for small values of~$\eps$). The convergence order of the existing $\eps$-uniformly convergent schemes does not exceed~1 in space and time. In this paper, using a defect correction technique we construct a special difference scheme that converges $\eps$-uniformly with the second (up to a logarithmic factor) order of accuracy with respect to~$x$ and with the second order of accuracy and higher with respect to~t. The conditions are given which ensure the $\eps$-uniform convergence of the defect-correction schemes with a rate of order $N^{-k}\ln^k N + K^{-k_0}}$, $k=1,2$, $k_0=1,2,3$, where $N+1$ and $K+1$ denote the number of the mesh points in $x$ and $t$ respectively. Theoretical results and the efficiency of the newly constructed schemes are confirmed with numerical experiments.