Mix parmesan cheese & garlic into sourdough bread dough before or after first rising?

If you are making sourdough you are working with natural yeasts in an acidic dough, so I wouldn't think adding parmesan and garlic is going to be a problem as long as you leave extra time for proofing. Parmesan is salty, and salt inhibits yeast. You are probably adding salt already so reducing the amount to compensate would make sense.

As for adding garlic in my experience it doesn't work that well just chopped up and in the bread whether you add it at the beginning or knead it in later. The flavor doesn't get out and you get chunks of intense garlic rather than a nice hum in every bite, although maybe that's the effect you want. I have a garlic and rosemary infused olive oil I use rather than adding garlic directly, another option is to put chopped garlic in olive oil (or butter, or any other oil) on very low heat for 10-15 minutes, this will infuse the oil with the garlic flavor which you add to the break. I haven't had any issues with this inhibiting yeast.

I would add these ingredients in at first mix rather than kneading it in later as kneading after first proof as you will knock out most of your air and damage the structure. You may need a longer proof but the structure will be better. Alternatively if your dough is stretchy enough (like a pizza dough or focaccia) you can stretch it flat after first rise, put ingredients in as a layer, then fold it over, then do your final rise and bake. That only works when you have a stretchy dough though.

My initial reaction is that adding cheese and garlic could affect the rise, unless you are using a recipe calibrated for their presence. Cheese would probably affect the texture of the bread, interfering with an even rise. Garlic is fairly astringent and has some antimicrobial properties; this could very well mess with the health of your yeast.

If you're going to try adding these, it would probably be best to knead them in rather than trying to add them during the first rise.

I've stuck all manner of ingredients into my dough during the mixing stage and never had any issue with rise or texture. Especially parmesan as it barely melts at all so you just end up with tasty lumps of cheese scattered through your final product. Cheddar on the other hand has a slight tendency to sink to the bottom and create a layer of cheese, personally I love it that way but I bet purists would not. The only thing I would suggest is adding mustard powder and dried onions flakes to help enhance the cheesy flavour as without it the bread masks and mutes much of the flavour.