They're two of the other major sport examples we have in the US, so feel free to find ones you think are more applicable.
Baseball does not have a significant parity problem compared to the other sports. Of the three sports I referenced:
The NBA has had 13 different teams win the championship since their soft cap began in 1985 (32.5%). There are seven teams with more than 2 championships during that time (Lakers 9, Bulls 6, Spurs 5, Warriors 4, Celtics 3, Pistons 3, Heat 3), which account for 82.5% of the total championships.
The NFL has had 15 teams win the Super Bowl since the salary cap began in 1994 (48.4%). There are three teams with more than 2 championships during that time (Patriots 6, Chiefs 3, Broncos 3), which account for 38.7% of the total championships.
MLB has had 14 different teams win the World Series since the luxury tax began in 1997 (50%). There are four teams with more than 2 championships during that time (Red Sox 4, Yankees 3, Giants 3, Dodgers 3), accounting for 46.4% of the total championships.
MLB has the the largest % of different teams win, and is in the middle of the three for 3+ championships won by a team. If anything it seems like more restrictive salary caps are harmful to parity, or at best don't make a difference.