In this paper, we present a parameter estimation analysis of the polarization and temperature power spectra from the second and third season of observations with the QUaD experiment. QUaD has for the first time detected multiple acoustic peaks in the E-mode polarization spectrum with high significance. Although QUaD-only parameter constraints are not competitive with previous results for the standard six-parameter ΛCDM cosmology, they do allow meaningful polarization-only parameter analyses for the first time. In a standard six-parameter ΛCDM analysis, we find the QUaD TT power spectrum to be in good agreement with previous results. However, the QUaD polarization data show some tension with ΛCDM. The origin of this 1σ–2σ tension remains unclear, and may point to new physics, residual systematics, or simple random chance. We also combine QUaD with the five-year WMAP data set and the SDSS luminous red galaxies 4th data release power spectrum, and extend our analysis to constrain individual isocurvature mode fractions, constraining cold dark matter density, αcdmi < 0.11 (95% confidence limit (CL)), neutrino density, αndi < 0.26 (95% CL), and neutrino velocity, αnvi < 0.23 (95% CL), modes. Our analysis sets a benchmark for future polarization experiments.