Yes. However, everyone I know who got gp alert/nhs text for dose 1 got it very much on schedule and booked for the next few days. The second dose is a worry because of the timing.
Would your parents be unhappy with using the book yourself in website? I've used it after my sister did it by chance and I passed it on to friends. However, my cev Dh and a couple of my friends want to wait to be asked. When I put my details in it recognised my nhs number and offered me early June and not before. It also confirmed, after I'd chosen, that it was my second dose.