If it is my opponent's turn and they move Irresistible Faefolk to Battlefield A and move my Tide Turner from Battlefield B to Battlefield A, can I ride my Tide Turner back to Battlefield B and score a point?

To answer your question: Yes, you can use Ride the Wind on your opponent's turn to move your Tideturner back to battlefield B and score a point, provided you meet the requirements for conquering that battlefield.

Sequence of Events

1. Opponent's Turn: Your opponent moves Irresistible Faefolk to Battlefield A (applying "Contested" status, making them the attacker).
2. Faefolk Trigger: The "When I move to a battlefield..." ability of Irresistible Faefolk triggers, and your opponent chooses to move your Tideturner from Battlefield B to Battlefield A.
3. Showdown Begins: Battlefield A is now Contested by opposing units, so a Combat Showdown begins.
4. Your Opportunity: During the Showdown, you have priority to play a [Reaction] or [Action] card. You play Ride the Wind (permitted by its [Action] keyword).
5. Ride the Wind Resolves: You move your Tideturner from Battlefield A back to Battlefield B.
6. Battlefield B Transition: Because your Tideturner is now present at Battlefield B, and assuming Battlefield B was empty/uncontrolled before you moved it there, Battlefield B becomes Contested.
7. Resolution and Scoring:
   • The original showdown at Battlefield A resolves.
   • Because your Tideturner is now at Battlefield B, a new, separate showdown triggers at Battlefield B after the current one concludes.
   • If you establish control of Battlefield B (e.g., you are the only player present at the end of the new showdown), you "Conquer" the battlefield.
   • Since you have not yet scored at Battlefield B this turn (the opponent's turn), you successfully gain a point.

Key Nuances

• "Once per Battlefield per turn": According to FAQ #825 and FAQ #2364, this restriction resets each turn, regardless of whose turn it is. Scoring at Battlefield B on your own turn does not prevent you from scoring at it again on your opponent's turn.
• Conquering Requirement: You only score if you actually establish control by conquering (Rule 464.1). If you merely moved to an empty battlefield and held it without a showdown occurring or without re-establishing control, you would not score. In this case, moving the Tideturner to an uncontrolled/empty battlefield will trigger a showdown, and winning/conquering that showdown allows you to score.
• Timing: Your Ride the Wind is played during the existing showdown at Battlefield A. It moves your unit to Battlefield B, which then triggers its own showdown process to determine control.

This is my interpretation based on the mechanics of Ride the Wind, Showdowns, and the "Once per battlefield per turn" scoring rule.