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The classification of supernovae (SNe) and its impact on our understanding of the explosion physics and progenitors have traditionally been based on the presence or absence of certain spectral features. However, current and upcoming wide-field time-domain surveys have increased the transient discovery rate far beyond our capacity to obtain even a single spectrum of each new event. We must therefore rely heavily on photometric classification, connecting SN light curves back to their spectroscopically defined classes. Here we present Superphot, an open-source Python implementation of the machine-learning classification algorithm of Villar et al., and apply it to 2315 previously unclassified transients from the Pan-STARRS1 Medium Deep Survey for which we obtained spectroscopic host-galaxy redshifts. Our classifier achieves an overall accuracy of 82%, with completenesses and purities of >80% for the best classes (SNe Ia and superluminous SNe). For the worst performing SN class (SNe Ibc), the completeness and purity fall to 37% and 21%, respectively. Our classifier provides 1257 newly classified SNe Ia, 521 SNe II, 298 SNe Ibc, 181 SNe IIn, and 58 SLSNe. These are among the largest uniformly observed samples of SNe available in the literature and will enable a wide range of statistical studies of each class.