Can my parents tell what free apps I've had on my phone?

My friend thought it would be funny to download a free dirty app on my phone. I didn't know it at the time, soni rattled off my password. When I got home later and was looking through the game he bought, I saw a "R-rated" one if you will. I quickly deleted it, but it was on my phone for at least an hour. I never plugged it into my computer, but will it be in my iTunes account? How can I delete it from there? I'm not afraid to tell my parents the truth, but I'd just like to avoid the situation if possible. They would have no reason to look through my history or anything, and they wouldn't find anything anyway... Except for this. Do they have eay access to this? Thanks!