We use 1169 Pan-STARRS supernovae (SNe) and 195 low-z (z<0.1) SNe Ia to measure cosmological parameters. Though most Pan-STARRS SNe lack spectroscopic classifications, in a previous paper we demonstrated that photometrically classified SNe can be used to infer unbiased cosmological parameters by using a Bayesian methodology that marginalizes over core-collapse (CC) SN contamination. Our sample contains nearly twice as many SNe as the largest previous SN Ia compilation. Combining SNe with cosmic microwave background (CMB) constraints from Planck, we measure the dark energy equation-of-state parameter w to be -0.989+/-0.057 (stat+sys). If w evolves with redshift as w(a)=w0+wa(1-a), we find w0=-0.912+/-0.149 and wa=-0.513+/-0.826. These results are consistent with cosmological parameters from the Joint Light-curve Analysis and the Pantheon sample. We try four different photometric classification priors for Pan-STARRS SNe and two alternate ways of modeling CC SN contamination, finding that no variant gives a w differing by more than 2% from the baseline measurement. The systematic uncertainty on w due to marginalizing over CC SN contamination, {sigma}_w_^CC^=0.012, is the third-smallest source of systematic uncertainty in this work. We find limited (1.6{sigma}) evidence for evolution of the SN color-luminosity relation with redshift, a possible systematic that could constitute a significant uncertainty in future high-z analyses. Our data provide one of the best current constraints on w, demonstrating that samples with ~5% CC SN contamination can give competitive cosmological constraints when the contaminating distribution is marginalized over in a Bayesian framework.