Tablet computers are widely used by young children. A report in 2016 shows that children aged 5 to 15 years are spending more time online than watching TV. A 2017 update of the same report shows that parents are becoming more concerned about their childrens online risks compared to the previous year. Parents are working hard to protect their childrens online safety. An increasing number of parents are setting up content filtering at home or having regular discussions with their children regarding online risks. However, although risks related to Social Media platforms or social video sharing sites (like YouTube) are widely known, risks posed by mobile applications or games (i.e. `apps) are less known. Behind the cute characters, apps used by children can not only have the possibility of exposing them to age-inappropriate content or excessive in-app promotions, but may also make a large amount of their personal information accessible to third-party online marketing and advertising industry. Such practices are not unique to childrens apps, but young children are probably less capable of resisting the resulting personalised advertisements and game promotions. In this report, we present findings from our online survey of 220 parents with children aged 6-10, mainly from the U.K. and other western countries, regarding their privacy concerns and expectations of their childrens use of mobile apps. Parents play a key role in childrens use of digital technology, especially for children under 10 years old. Recent reports have highlighted parents lack of sufficient support for choosing appropriate digital content for their children. Our report sheds some initial light on parents key struggles and points to immediate steps and possible areas of future development.