If I held a Battlefield from my turn and passed my turn, the Battlefield was then taken over by my opponent and they scored a point, can I Ride the Wind a unit to conquer and gain another point from that Battlefield?

Yes, you can use Ride the Wind to move a unit to a battlefield, conquer it, and score a point, even if you already scored at that battlefield during your own turn.

Ruling

According to FAQ #2364 and FAQ #5175, the restriction "once per battlefield per turn" refers to each player's individual turn. Scoring at a battlefield during your own turn does not prevent you from scoring at that same battlefield again during your opponent's turn.

Sequence

1. Your Turn: You held the battlefield and scored.
2. Opponent's Turn: Your opponent moves a unit into the battlefield, triggering a showdown.
3. Your Opportunity: During the showdown on your opponent's turn, you play Ride the Wind to move your unit into that battlefield.
4. Resolution: If you win the resulting combat showdown, you establish control of the battlefield. Because the battlefield transitioned from being under your opponent's control to yours, you conquer it.
5. Scoring: Since you have not yet scored at that battlefield during this specific turn (the opponent's turn), you successfully score a point.

Important Nuances

• Control Requirement: You must completely lose control of the battlefield and have the showdown end (or have the battlefield become uncontrolled/contested) before you can score it again. If you maintain control of the battlefield throughout the entire showdown (even if your unit is removed and replaced during that same showdown), you will not score when you win (see FAQ #1067).
• Final Point Restriction: As noted in FAQ #7467, if this would be the final point of the game, you cannot score it unless you have already scored at every battlefield "this" turn. If you have not, you would draw an extra card instead of gaining the final point.

This interpretation is based on the provided FAQs (#2364, #5175, #1067, #7467) and Rule 447, which confirms the "once per battlefield per turn" restriction resets with each new turn.

Note: I could not fully verify this ruling against the retrieved rules.