In this article we present the level of knowledge and literacy held by the parents of primary school students regarding internet safety (online safety, digital safety) in the context of digital literacy (DL) in terms of both technical skills and knowledge. The study reported on here was conducted in Poland, and was commissioned by the Ministry of National Education. The research involved measuring the knowledge and skills regarding the prevention of electronic threats (e-threats) which are defined as problematic situations and behaviour mediated by digital media and the internet. E-threats are related to mental and physical health, social aspects and technical matters related to the use of information and communication technologies (ICTs). In order to narrow down the research area, 6 components related to e-threats were selected, namely: the ergonomics of using digital media; the evaluation of the reliability of online information; the influence of advertising on consumers' choices; risky online relationships; understanding the mechanisms of secure logging-in; and protection against malware. The DL level was measured using a competence test. An analysis of 514 tests revealed that the most neglected areas were the protection of children against unwanted communication with other users, and secure logging in. The strongest component was the ergonomics of using ICT. We also noted that more than one fifth of parents showed a satisfactory level of DL related to online safety. Another observation was that DL is a complex and heterogeneous concept. The DL components are related with one another to different extents. Some DL elements are determined by the level of education, place of living, subjective sense of own well-being or intuitive perception (self-evaluation).