Hi Laura,
The Diyonisos Hotel is right in the centre of Kato Paphos opposite a really good cocktail bar called The Flintstones ( we go most nights!) Your parents will be right in the midst of things but whenever we have been into the hotel itself it doesn't seem too noisy and it always seems to have plenty of people staying there so it can't be too bad. You are only a couple of minutes walk away from the sea front and the main road where you/they can catch a bus into town or over to Coral Bay or the next one along Corallia where the swimming and beaches are great. Hope they have a good time, all the best Amber and Karl