In shared bicycle/pedestrian facilities next to a road, do I obey the pedestrian or the vehicle traffic control devices?
Honestly, I think that's a crappy design for a bike path, and I would just ride in the road instead.  A bike path like that is great for a weekend ride, but for a daily commute, it just wouldn't cut it.

If that road isn't safe to ride in, I'd try to find a different route.  And if that's not an option, I suppose I would probably ride against the pedestrian light, slowing down to make sure it's safe to do so.