What does v. mean in marriage? For example: Klive Arthur Parry v. Kenneth Stark?

It usually means a divorce. It stands for "versus"

It means one has filed legal action against the other one so they are on different sides challenging each other over something.

You might see the same thing in the sport of a boxing or wrestling advertisement.

Smith v. Jones meaning they are fighting each other. Challenging each other usually for a title or on the road to earn enough wins to qualify to fight for the title.

Sometimes you may see it represented as vs. but it means the same thing.

Smith vs. Jones