Red Cards issued during regular time in Extra Time

No. There's only one theoretical edge case in which a team with ten players could be allowed to even up the numbers again in extra time:

1. A team uses up all their substitutions
2. They then have a player come off injured, forcing them to play with ten players since they can't substitute on a fit one
3. The tournament allows one extra substitution in the case of extra time (e.g. Euro 2020)

In that scenario, they could theoretically (I'm not sure whether this is actually allowed in practice) bring on a fit player to replace the injured one and get back up to eleven men.

In all other scenarios, the team with ten players would be forced to play on with ten, and their opponents would keep their numerical advantage (i.e. they wouldn't be forced to bring someone off to make it 10 v 10). As an example, the Euro 2020 match between Spain and Switzerland just finished, and the teams' numbers were not equalised in extra time.