This job that you offered means a lot to me, so I told my parents that I would be unable to go on vacation with them.
Say something to that extent, but word it better, and it will look like you are comitted to working where you want to work. As for you not being able to work a certain night at a certain time, just say that you have a prior commitment that you cannot miss. If she asks what it is, make something up, but make it up in advance and practice it so that it sounds like the truth.
Either that, or just tell her that you already promised your dad that you would help him out with something this one time only. This way, it looks like you are comitted, and that you won't be working with your dad again, as to show that it will not interfere with your Walgreen's schedule.