To reduce the transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), most countries closed schools, despite uncertainty if school closures are an effective containment measure. At the onset of the pandemic, Swedish upper-secondary schools moved to online instruction, while lower-secondary schools remained open. This allows for a comparison of parents and teachers differently exposed to open and closed schools, but otherwise facing similar conditions. Leveraging rich Swedish register data, we connect all students and teachers in Sweden to their families and study the impact of moving to online instruction on the incidence of SARS-CoV-2 and COVID-19. We find that, among parents, exposure to open rather than closed schools resulted in a small increase in PCR-confirmed infections (odds ratio [OR] 1.17; 95% CI [CI95] 1.03 to 1.32). Among lower-secondary teachers, the infection rate doubled relative to upper-secondary teachers (OR 2.01; CI95 1.52 to 2.67). This spilled over to the partners of lower-secondary teachers, who had a higher infection rate than their upper-secondary counterparts (OR 1.29; CI95 1.00 to 1.67). When analyzing COVID-19 diagnoses from healthcare visits and the incidence of severe health outcomes, results are similar for teachers, but weaker for parents and teachers' partners. The results for parents indicate that keeping lower-secondary schools open had minor consequences for the overall transmission of SARS-CoV-2 in society. The results for teachers suggest that measures to protect teachers could be considered.