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Abstract Photometric classifications of supernova (SN) light curves have become necessary to utilize the full potential of large samples of observations obtained from wide-field photometric surveys, such as the Zwicky Transient Facility (ZTF) and the Vera C. Rubin Observatory. Here, we present a photometric classifier for SN light curves that does not rely on redshift information and still maintains comparable accuracy to redshift-dependent classifiers. Our new package, Superphot+, uses a parametric model to extract meaningful features from multiband SN light curves. We train a gradient-boosted machine with fit parameters from 6061 ZTF SNe that pass data quality cuts and are spectroscopically classified as one of five classes: SN Ia, SN II, SN Ib/c, SN IIn, and SLSN-I. Without redshift information, our classifier yields a class-averagedF₁-score of 0.61 ± 0.02 and a total accuracy of 0.83 ± 0.01. Including redshift information improves these metrics to 0.71 ± 0.02 and 0.88 ± 0.01, respectively. We assign new class probabilities to 3558 ZTF transients that show SN-like characteristics (based on the ALeRCE Broker light-curve and stamp classifiers) but lack spectroscopic classifications. Finally, we compare our predicted SN labels with those generated by the ALeRCE light-curve classifier, finding that the two classifiers agree on photometric labels for 82% ± 2% of light curves with spectroscopic labels and 72% ± 0% of light curves without spectroscopic labels. Superphot+ is currently classifying ZTF SNe in real time via the ANTARES Broker, and is designed for simple adaptation to six-band Rubin light curves in the future.