It would be true (and grammatical) for you to say both that:
If it hadn't been for my parents
....I wouldn't HAVE BEEN born
referring to an action in the past that marked the start of your life
....I wouldn't BE living now
referring to the continuing consequence of your having been born.
Note the changes to your examples.
Although you might have said:
I wouldn't live now
The continuous form (be living now) is more natural in this context as you are still living.
In general, we use the present continuous to talk about what we are doing now/today/soon/in the future, as in:
I am going to the shops/going swimming/going home/going to France next year
We use the simple present to talk about our habits, routines:
I go to the shops/swim/practise twice a week.