Looking at the cyber sitter site and the FAQ, etc. they don't seen to give much information on how they achieve the filtering.
It seems to me that their recommendation is to disabled other products rather than actually resolve the problem as more and more AVs are going to be monitoring HTTP traffic for malware. Not to mention recommend not using certain products at all for supposed non-communication and recommend an AV product that personally I feel doesn't provide the same level of protection as avast.
For it to say they have communicated with avast support with no response, flies in the face of what is usual for avast, as can be seen in many such topics about conflicts with other software. It just doesn't compare with these other instances.
Whilst checking this out I did see one area which did a comparison with other parental control software and cyber sitter didn't compare that well with other products.