Buckingham Palace declined to comment on the vandalism incident.
Fathers 4 Justice emerged about a decade ago and made headlines with high-profile demonstrations, including having members dressed as superheroes scale Buckingham Palace and Westminster Abbey.
In 2004, members caused panic in the House of Commons chamber when they hurled purple powder at then-Prime Minister Tony Blair from the public gallery.
The original group splintered in 2006 following allegations that extremist members were plotting to kidnap Blair's young son Leo.
Stanesby — who was once fined for climbing the Stonehenge monument dressed as cartoon character Fred Flintstone — said the group had spent several years trying to lobby politicians for changes in the law. He said it had been ignored by politicians and would now revert to "nonviolent direct action."