Two-hundred preschool children in Head Start (55% girls; 20% Hispanic, 25% African-American, 55% European American; [mean] age = 4.80 years old) participated in a randomized-controlled trial of a home visiting intervention designed to promote emergent literacy skills (the Research-based Developmentally Informed parent [REDI-P] program). This study explored concurrent changes in levels of parent support and child literacy skills that occurred over the course of the intervention, and examined the impact of pre-intervention parent support and child literacy skills as potential moderators of parent and child outcomes. Cross-lagged structural equation models and follow-up analyses indicated that intervention had the strongest impact on child literacy skills when parents were high on support at the pre-intervention assessment. Conversely, the REDI-Parent program promoted the greatest gains in parent support when parents entered the program with low levels of support. These findings suggest that families may benefit from home visiting school readiness interventions in different ways: child skill acquisition may be greatest when parents are initially high in support, whereas parenting may improve most when parents are initially low in support. (author abstract)