I think the answer depends on where you want to go and where you are coming from. Are you a hugging type of person with your friends or are you generally not a touchy/feely kind of person? I would suggest starting off slowly with simple thanks and do it repeatedly, consciously aware that you are getting yourself used to these types of behaviors with your parents. Set a small goal ahead of where you are now and work towards achieving it – maybe giving your dad a handshake if you’re a guy or squeezing your mom’s arm if you’re a girl (not saying that girls can’t touch dads and boys can’t touch moms, only that touching the parent of the same sex is probably easier to start with). You can make incremental changes in how you speak and behave physically.
Regarding a letter, that may be a good idea for yourself so that you can read it and realize how much you have to be grateful to and have appreciation for your parents. Building up the hakores hatov in your heart may make it easier for you to express it verbally and/or physically.