I live so close to some amazing trails riding ( bottom of burke/eagle )
While there are some trails on eagle that have been built for motos, as far as I know there are none on Burke. I, among others, actively maintain the mtb trails on Burke and Eagle. When motos ride the mtb trails it really sets us back on doing repairs and keeping the trails running smoothly. I know there has been some recent sightings and run ins with someone riding a Yamaha on the Burke trails. I really hope this isn't you as one who frequents a mtb forum should know the amount of effort and work that goes into maintaining trails. I too am a Moto and trials guy and there are more than enough riding areas that this really isn't needed. Mcnutt and ioco have tons and tons of trails available to us and eagle is also getting a small collection of trails going too. While the eagle Moto trails are pretty rough for the average rider, they are more manageable on a trials bike. While there are some issues with motos on mtb's trails on eagle the two user groups are trying to work things out and hopefully with some good signage it will keep things in check.
As far as I am aware there is no motorcycling in provincial parks.
I know on one of the trails that my dad, wife and I maintain on Burke there was a lot of damage done to it recently. This will involve many hours to repair.