Is it permissible to supplicate while performing a righteous deed, such as honoring one's parents or going to the mosque, and is this better?

The question is unclear. However, there is no objection to supplicating (making du'a) during righteous deeds. The Sunnah has indicated its permissibility during circumambulation (Tawaf), between Safa and Marwah, at Arafat, at Al-Mash'ar Al-Haram, and when going to the mosque. One must be careful not to specify a particular supplication that has not been mentioned in the Sharia, so as not to fall into innovation (bid'ah).