We investigate the existence and especially the linear stability of single and multiple-charge quantized vortex states of nonlinear Schroedinger equations in the presence of a periodic and a parabolic potential in two spatial dimensions. The study is motivated by the examination of pancake-shaped Bose-Einstein condensates in the presence of magnetic and optical confiement. A two-parameter space of the condensates chemical potential versus the periodic potentials strength is scanned for both single- and double-quantized vortex states located at a local minimum or a local maximum of the lattice. Triply charged vortices are also briefly discussed. Single-charged vortices are found to be stable for cosinusoidal potentials and unstable for sinusoidal ones above a critical strength. Higher charge vortices are more unstable for both types of potentials and their dynamical evolution leads to breakup into single-charged vortices.
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