SARS-CoV-2 transmission dynamics within households involving children are complex. We examined the association between paediatric index case (PIC) age and subsequent household SARS-CoV-2 transmission among cases reported to the Minnesota Department of Health between March 2021 and February 2022. In our primary analysis, we used logistic regression to estimate odds ratios adjusted for race/ethnicity, sex, geographic region, and disease severity among households with an unvaccinated PIC. We performed a secondary analysis among households where the PIC was eligible for vaccination adjusting for the same covariates plus time since the last vaccination. Both analyses were stratified by variant wave. During the Alpha wave, PICs of all age groups had similar odds of subsequent transmission. During Delta and Omicron waves, PICs aged 16-17 had higher odds of subsequent transmission than PICs aged 0-4 (Delta OR, 1.32; [95% CI, 1.16-1.51], Omicron OR, 4.21; [95% CI, 3.25-5.45]). In the secondary analysis, unvaccinated PICs had higher odds of subsequent transmission than vaccinated PICs (Delta OR 2.89 [95% CI, 2.18-3.84], Omicron OR 1.35 [95% CI, 1.21-1.50]). Enhanced preventative measures, especially for 12-17-year-olds, may limit SARS-CoV-2 transmission within households involving children.