No.

Great, thanks! What about the parging on the outside, it looks like a thin layer of concrete too?

So even if it did, no big deal. Ventilate and PPE.

No, again.

If you are worried about it, get it tested.

Do you realize that asbestos is everywhere?

We are all exposed to low levels of asbestos in the air we breathe every day. Ambient, or background, air usually contains between 10 and 200 asbestos fibres in every 1000 litres (or cubic metre) of air (equivalent to 0.01 to 0.20 fibres per litre of air) meaning we may breath up to 5000 fibres per day.

Helpful? lol

I called a testing place today, they said don't bother testing the actual foundation but some parging materials can contain it. So I took in a sample.

Cilantro, I do know there is background asbestos. But I'm clueless and paranoid and I don't want to make any big mistakes that I'll regret later!

Regardless of whether or not there is asbestos in the parging, you wear PPE because of the silica dust. 100% guaranteed to be releasing a cloud of silica dust in both the foundation & parging. Two birds/ one stone.

Our contractor is going to do the actual drilling. It's my kids living in the space afterward that I'm concerned about. If we HEPA vacuum and mop the room after, would that make it safe for us?

Yes concrete does not contain asbestos and as you alreay caaried out the testing so i think you will be ok.