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Most molecular diagram parsers recover chemical structure from raster images (e.g., PNGs). However, many PDFs include commands giving explicit locations and shapes for characters, lines, and polygons. We present a new parser that uses these born-digital PDF primitives as input. The parsing model is fast and accurate, and does not require GPUs, Optical Character Recognition (OCR), or vectorization. We use the parser to annotate raster images and then train a new multi-task neural network for recognizing molecules in raster images.We evaluate our parsers using SMILES and standard benchmarks, along with a novel evaluation protocol comparing molecular graphs directly that supports automatic error compilation and reveals errors missed by SMILES-based evaluation. On the synthetic USPTObenchmark, our born-digital parser obtains a recognition rate of 98.4% (1% higher than previous models) and our relatively simple neural parser for raster images obtains a rate of 85% using less training data than existing neural approaches (thousands vs. millions of molecules).