Can a parent prevent someone else taking their children out?

Answer yes.

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What is the penalty for one parent taking children out of the country without the other parent's consent?

Whether this is even a crime depends on which country the children are taken from and to, and whether one or both of the parents have legal custody of the children.

Can a non-custodial parent prevent a custodial parent from Homeschooling children in Washington state?

It's up to a judge, but most likely he would just order regular reviews. see link below

If you live in Texas with joint custody can one parent take the child out of the country to visit relatives without consent of the other parent and should you prevent this?

Answer No, When a child under the age of 18 is traveling outside the U.S. with only one parent or a third party there must be notarized written permission from the absent parent(s) or legal guardia... Read More »

Does a divorced parent need a notarized statement from the other parent to fly out of state with children?

Answer That would depend on whether the divorce decree stated Joint Custody or not, and probably what state you live in.