If I have conservatorship over my parents, does this trump my sisters position of trustee over their rental property?

Like I said, two different scenarios. Being your parents' conservator is totally and legally separate from you being a beneficiary of a Trust.

Well, I need the income from the rental property to pay for their care and I am the beneficiary of the rental trust.

No, because neither one of you is playing in the same card game. You have totally separate duties.