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You are looking for TransformedRegion. GeometricTransformation is for transforming graphics primitives, but you are looking at region functionality. This is a case where the difference is important. Simply, t = TransformedRegion[p, AffineTransform@A] (* Parallelogram[{0, 0}, {{-1, 2}, {3, 0}}] *) where the AffineTransform was needed as TransformedRegion ...

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