For me, I only have my cell, haven't had a landline in probably 8 years. It's always with me especially when we go outside or for a walk. I MUCH prefer texts and emails because I can answer them when I want. I HATE phone calls lol. I am terrible for screening my calls lol.

I haven't had an issue with parents who needed constant reassurance (I expect multiple texts for the first couple weeks but after that it seems to stop) but if I was finding parents were doing this more and more OR I was getting annoyed with it, I would write up a little blurb about communication in my contract so that parents know from the get go what I expect and will tolerate.

Be honest with the parents if this is something that is truly annoying you and keeping you from taking care of the kids. I am a huge multi-tasker so I don't find texts annoying but I definitely can see why some would be frustrated with it! Don't stress about something that can easily be fixed Good luck!