Parent ego state really is a state where the emotions don’t reside (or to a much lesser extent than the Child ego state).
There is a difference between being self-conscious and self-aware. Being self-conscious means that my emotions tend to take over, whereas self-awareness does not include emotions but it does still allow me to observe them.
I notice that if I start to become self-conscious then all I need to do is to be with myself, observe my breathing, my thoughts and my emotions. This is an effective way of calming and centering myself. When I do so, the reasons for my self-consciousness dissipate or resolve themselves without my intervention.