I think you and your partner need to follow your own desire to have a quiet wedding with no fuss. The only way you will get no fuss is if you tel no one else and invite no one.
If you tell both sets of parents beforehand there may be a desire for gifts, cards, a celebratory meal etc, and they will tell the rest of your siblings so it will create waves.
Whereas afterwards they can’t do much except congratulate you. And you can always say you got married yesterday, or last week, or last year or 2015. No one needs to know.
I would be a bit sad if my child got married in secret but I would get over it in a few minutes. I would however be very hurt and upset if some relatives on either side had been in on the secret or even attended the wedding, while all the time I was in ignorance.
I had a girl who I worked with announce once that she had gotten married in secret 3 years previously. She made a big song and dance about it, showed us photos of the wedding etc. I just found it strange, her making a big fuss of it 3 years later. If she had mentioned in passing she was married a few years I would not have blinked an eye, and just carried on.
We had another good friend couple called to our door one day, guess what? Engaged? No married. We went to the pub and had a drink to celebrate with them. It wasn’t a big deal, they just did what suited them.