Funny thing I found was that outside of our hardcore drinking buddies people don't seem to care that much if I decide to have a drink or not. Well, some regular friends have also been uncomfortable but I know that is because they also have some challenges with alcohol & me not drinking makes them more aware of their problem/uncomfortable themselves.
In regards to your parents Vlad & the others have given you a good list. I personally have gone with "I am not drinking for health reasons" and my family members were fine with that... then again they have all seem me drunk at one point or another ;-)
There is nothing wrong with "I am trying to be healthier", thats the truth isnt it? If they question further you can take it from there & gauge where they/you are at.
You will be fine as long as you dont pick up that first drink
Let us know how it goes.
Cheers ~ NB