Parental and familial factors influence numerous aspects of adolescents’ lives, including their physical activity level (PAL). The purpose of this study was to evaluate the changes in PAL which occurred during the COVID-19 pandemic, and to evaluate influence of sociodemographic and parental/familial factors on PAL levels before and during pandemic in adolescents from Bosnia and Herzegovina. The sample included 688 adolescents (15–18 years of age; 322 females) who were tested on two occasions: in January 2020 (baseline; before the COVID-19 pandemic) and in April 2020 (follow-up; during the COVID-19 pandemic lockdown). Variables included PAL (measured by the Physical Activity Questionnaire for Adolescents–PAQ-A) as well as sociodemographic-, parental-, and familial factors. A significant decline in PALs was recorded between baseline and follow-up (t-test: 11.88, p < 0.001). Approximately 50% of adolescents underwent sufficient PAL at baseline, while only 24% of them were achieving sufficient PAL at the time of follow-up measurement. Paternal education was positively correlated (OR (95%CI): baseline: 6.63 (4.58–9.96), follow-up: 3.33 (1.19–7.01)), while familial conflict was negatively correlated (baseline: 0.72 (0.57–0.90), follow-up: 0.77 (0.60–0.99)) with PALs before and during the pandemic. This study highlights the importance of the parent–child relationship and parental/familiar support in promoting physical activity both during regular life and during crises and health challenging situations like the COVID-19 pandemic.