Do algorithms for solving parity games ignore other possible strategies for player V after finding one?
Parity games can be reduced to discounted payoff games. See, e.g.,

Marcin Jurdziński (1998). Deciding the winner in parity games is in UP ∩ co-UP. Information Processing Letters 68:3 119-124.

As explained in that paper (and first observed by Zwick and Paterson, I believe), there are min/max optimality equations for discounted payoff games with a unique solution that assigns a "value" to each state. From these values, it is easy to read off all optimal strategies of the players.

Discounted payoff games can in turn can be solved as a P-matrix Linear Complementarity Problem (LCP), where the reduction uses the optimality equations and from a solution to the LCP one could again read off all optimal strategies. See,

Marcin Jurdziński and Rahul Savani (2008). A simple P-matrix linear complementarity problem for discounted games. Computability in Europe (CiE), 283-293.

Now in the following paper a discrete strategy improvement algorithm for parity games is given that essentially defines and uses optimality equations.

Jens Vöge and Marcin Jurdziński (2000). A Discrete Strategy Improvement Algorithm for Solving Parity Games. Computer-aided Verification (CAV).

It's very possible you could read off all strategies from a "solution" to those optimality equations, e.g., obtained via their algorithms - but, I am not sure.