We cannot 'blame' your parents for your current temperament or behavior. Your parents were just being who they were, acting like they acted, because it is how they expressed themselves. It is what it is.
The natal Saturn in your 4th of home and childhood does tell us that you were possibly left with some painful residue, which may have you feeling emotionally stuck in old childhood patterns. Feelings of not living up to lofty expectations, and fears from the past? Or feeling of not measuring up , even if it is just a fear and not a reality.
Your Progressed Sun tells me that you have the power and free will and the ability to modify and improve upon your current way of being and doing. It is up to you.
Studying your birth chart is a very good way to help you free up old patterns and create better ways of coping. I am glad you are doing that.
Your Progressed Sun has finally moved into your 3rd house of self expression and communication. You are now learning how to better express yourself socially and in the community. The 3rd house opens up your world and allows you to let a few more people in. You will feel more social, more welcoming and more comfortable in those kinds of social situations. You will be more confident meeting new people and making new friends.
Also, the Sun will progress into Scorpio, transforming your emotional temperament as well. You will be less 'critical' and detached and will likely be more emotionally connected and have feelings that you have not experienced before.