Okay, I might be a little late in this, but lemme get this straight.

You have a history of pathological lying and manipulation. Your parents no longer know when to believe you, and when you are being lying and manipulative. You caused the dog to yelp out because you were being overly rough with it, and they understandably were upset because the dog yelped in pain. You claim they said you were abusing the dog, when you claim you were just being absentmindedly overly rough.

Yeah, I'd side with the parents on this one.

However, to prevent this from happening, you need to stop the pathological lying and manipulation and make an honest effort to build up their trust in you again. It's basically on you that you their trust in your honesty is null at this point, so it's on you to work hard to prove to them that they can trust you again.