The phrase 'a heck of a lot' means 'very much'. It's a euphemism for 'a hell of a lot'. It's the same as 'a lot' but when you add the 'heck of' or 'hell of' it means more than just 'a lot'.
The phrase 'to live up to' is talking about meeting a high standard. When there is a standard level of achievement that is to be met we can say we will 'live up to the standard'. This person's mother and father set a high standard of achievement for their children to meet, if they want to follow their parents' example. The children have to 'live up to' that standard and it's a 'heck of a lot' for them to do.
Another difficult part of this phrase to understand is why the verb is 'give' and not something else. Why is it not
You set us a heck of a lot to live up to.
You require us to live up to a heck of a lot.
which is what you'd expect if you're comparing it to 'set a high standard to live up to'?
This is mostly just idiomatic. This phrase is a standard way of expressing the thought. It follows a pattern that is used in many different situations.
You give me something to live up to.
You give me great inspiration.
You give me something to hope for.
She gives me someone to look up to.
He gives me a sense of dread.
The sense of meaning here as that the person who is doing the giving does it without being aware of doing it. They give just because of who they are and how they live their lives. They are not trying to set a standard, or provide hope or be inspirational, or be creepy. It happens automatically, without their making any extra effort.