Multi-employer pension plans are usually governed by a board of trustees. Depends on the plan but typically labor and management share power equally (broadly speaking).
And yeah, pensions, healthcare are all part of collective bargaining. Do you control the healthcare plan where you work? I assume it's part of negotiating the terms of your employment, so you could always ask for a really good one.
But even if your union doesn't directly control your healthcare plan, I would still assume it was part of the collective bargaining?