Parental vaccine hesitancy could be the next frontier in tackling COVID-19. There is a very real risk that even already-vaccinated parents refuse to vaccinate their children, thereby allowing the virus to persist and spread in classrooms, playgrounds and beyond. To better understand this problem we administered a survey on a sample of 1,549 UK parents who are broadly representative of the UK in terms of gender and ethnicity. We find that parents are generally slightly more concerned about vaccinating their children than vaccinating themselves. 26% of parents said that they will either definitely not, or are very unlikely to, vaccinate their children. 20% of parents thought that vaccines were less safe for their children than for themselves, and parents thought that vaccines were more safe for their children in only 8% of cases. We also examined factors associated with parents who are unlikely to vaccinate their children. The most important factor is whether the parent has already been vaccinated or is likely to get vaccinated. We use the ‘3Cs’ model (Confidence, Convenience and Complacency), from SAGE and WHO, to understand other drivers of parental vaccine hesitancy. To our knowledge, this paper is the first to use the model to understand parental vaccine concerns. We identify a range of demographic, social and individual factors which are associated with whether parents are likely to vaccinate their children.