An electronic device may have an antenna embedded in a substrate. The substrate may have first layers, second layers on the first layers, and third layers on the second layers. The antenna may include a first patch on the first layers that radiates in a first band, a second patch on the second antenna layers that radiates in a second band, and a parasitic patch on the third layers. A short path may couple ground to a location on the first patch that allows the first patch to form a ground extension in the second band for the second patch without affecting performance of the first patch in the first band. The first layers may have a higher dielectric permittivity than the second and third layers to minimize the thickness of the substrate without requiring a separate dielectric loading layer over the substrate.