Various innovations facilitate the use of intra-picture prediction modes such as palette prediction mode, intra block copy mode, intra line copy mode and intra string copy mode by an encoder or decoder when wavefront parallel processing (“WPP”) is enabled. For example, for a palette coding/decoding mode, an encoder or decoder predicts a palette for an initial unit in a current WPP row of a picture using previous palette data from a previous unit in a previous WPP row of the picture. Or, as another example, for an intra copy mode (e.g., intra block copy mode, intra string copy mode, intra line copy mode), an encoder enforces one or more constraints attributable to the WPP, or a decoder receives and decodes encoded data that satisfies one or more constraints attributable to WPP.