Maternal dietary quality during pregnancy is associated with offspring outcomes. These associations have not been examined in three-generation families. We investigated associations between parental and grandparental dietary quality, determined by healthy eating index (HEI)-2015, and offspring birth outcomes and weight status at age 5. The Lifeways cohort study in the Republic of Ireland comprises 1082 index-child’s mothers, 333 index-child’s fathers, and 707 grandparents. HEI-2015 scores were generated for all adults from prenatal dietary information collected using a validated food frequency questionnaire. In an adjusted model, greater adherence to the maternal HEI was associated with lower likelihood of low birth weight (LBW) (OR: 0.72, 95% CI: 0.50–0.99, p = 0.04). Similarly, maternal grandmothers (MGM) with higher HEI scores were less likely to have grandchildren with LBW (OR: 0.87, 95% CI: 0.61–0.96, p = 0.04) and more likely to have macrosomia (OR: 1.10, 95% CI: 1.01–1.22, p = 0.03). Higher paternal and paternal grandmothers (PGM) HEI scores were associated with lower likelihood of childhood obesity (OR: 0.89, 95% CI: 0.30–0.94, p = 0.03) and overweight (OR: 0.83, 95% CI: 0.22–0.99, p = 0.04), respectively. Mediation analysis showed significant direct relationship of MGM and PGM HEI scores on grandchildren’s birthweight and obesity, respectively. In conclusion, maternal line dietary quality appears to influence fetal growth whereas paternal line dietary quality appears to influence postnatal growth.