There are many manners of social interaction (ādāb) which one learns from books and there are others which are learnt directly from teachers or elders. In both there is great wisdom and application.
I remember being told that when walking with a scholar or elder (e.g. a parent) that one should stand to their right, just behind them. Not really walking in front of them, or directly behind. Just two reasons I want to point out;
1) It’s mentioned that the great ḥanbali jurist of his time, al-Qāḍi Abu Ya’la (ra) said to his student al-Karkhī ‘If you walked with someone you respect, where would you walk?’. His student said ‘I don’t know’. Abu Ya’la said ‘Walk to his right and place him at the position of the Imam in the prayer’.
I see from this the concept of respecting one’s teacher/parent, the same way we give reverence to the Imām leading us in prayer. Yes, it’s not worship, but it’s about showing respect.
2) The other reason, is that by being behind on the right side, this is the best place to be ‘in service’. If the person of honour has his hands full, he can pass something to you with ease. If they need you, they don’t have to look behind or rush to catch up with you. If they need to speak to you, you are within close distance without having to shout etc.
Yes, we may not find direct evidences from the Qur’ān and sunnah, but much is to be learnt from the interactions with scholars as they learnt from their teachers the embodiment of the sunnah.