Many risk behaviours in adolescence are socially patterned. However, it is unclear to what extent socioeconomic position (SEP) influences adolescent drinking in various parts of Europe. We examined how alcohol consumption is associated with parental SEP and adolescents' own SEP among students aged 14-17 years. Cross-sectional data were collected in the 2013 SILNE study. Participants were 8705 students aged 14-17 years from 6 European cities. The dependent variable was weekly binge drinking. Main independent variables were parental SEP (parental education level and family affluence) and adolescents' own SEP (student weekly income and academic achievement). Multilevel Poisson regression models with robust variance and random intercept were fitted to estimate the association between adolescent drinking and SEP. Prevalence of weekly binge drinking was 4.2% (95%CI = 3.8-4.6). Weekly binge drinking was not associated with parental education or family affluence. However, weekly binge drinking was less prevalent in adolescents with high academic achievement than those with low achievement (PR = 0.34; 95%CI = 0.14-0.87), and more prevalent in adolescents with >€50 weekly income compared to those with ≤€5/week (PR = 3.14; 95%CI = 2.23-4.42). These associations were found to vary according to country, but not according to gender or age group. Across the six European cities, adolescent drinking was associated with adolescents' own SEP, but not with parental SEP. Socio-economic inequalities in adolescent drinking seem to stem from adolescents' own situation rather than that of their family. The online version of this article (doi:10.1186/s12889-017-4635-7) contains supplementary material, which is available to authorized users.