“My parents set really high standards for me, and I haven't always lived up to them.”

You're having a deep talk with a friend about how you were raised. When you were young, your parents pushed you hard to be successful. But you don't feel like you're very successful, and sometimes you don't work very hard. You say:

My parents set really high standards for me, and I haven't always lived up to them.

set standards for (someone/something)

A "standard" is a level that you decide that something should be equal to or greater than. For example, if you expect your children to get a "B" grade or better in their classes, that's a "standard" that you have for them. You describe standards as being "high" or "low":

I love how their standards for quality are so high.

He's always had low standards when it comes to the opposite sex. I mean, that guy will sleep with anybody.

To "set" a standard for something means to decide what the standard is going to be:

As team leader, it's your job to set a standard for the entire group.

As team leader, it's your job to set a standard so

Setting a standard is similar to "setting" the rules for something:

I'm sorry. I don't set the rules; I just enforce them.

live up to (someone's) standards

To "meet" a standard means to reach the level that has been chosen for something:

Teachers who fail to meet the new standards will be suspended or fired.

The phrase "live up to (a standard)" means to reach a standard that is high. This phrase also has a more emotional feeling than "meet a standard".