Am I contacting my parents too often if I email them on average several times a day? Does that indicate that I’m not being independent? I am generally the one initiating the conversations, not them.
There is no “normal”. Most parents would be thrilled for this issue. Maybe you will pull away, maybe you won’t. There is no “wrong”…can you ask my kid to at least answer my texts? 
My 20-year-old son sends short texts, sometimes several times a day. I like hearing what he’s up to! If I respond, I keep it short.
I still talk to my parents several times a week, and I’m old! It’s nice that you have such a good relationship with your folks.
I talk to my mom pretty much every day through Facebook. Usually it’s us sharing short stories about our day and commenting on things that are going on with family/friends/housemates/classes/work/etc, but a lot of times it goes more into detail if we both happen to have more free time. I’ve always been close to my mom though; I consider her one of my best friends. We just got closer after I went to college. Likewise, she calls her mom every week or so and they talk for a couple hours at a time.
Just to give a couple comparisons, consider my housemates. One of them calls her parents and brother at least twice per day: Once in the morning and once in the evening. The other one talks to her older brother via Skype almost every day but only talks to her parents once every week or so. Both have healthy relationships with their families, but different family dynamics.
If you’re all comfortable with it, I don’t see a reason to stop. If you’re not comfortable with it, it’s your decision to cut back a bit. You’re an adult now, so it’s up to you to decide exactly how much contact you want to have. Likewise, it’s nobody else’s business how much you talk to your parents.
There is no right or wrong answer. Unless you are talking to parents and taking away from socializing with people you’re fine. I text my mom daily and sometimes its just to see how she’s doing