Personally I wouldn't show my parents videos of freeleethebananagirl if for no other reason than her way of eating is undoubtedly alarming to the general public. Not all vegans live on just pounds of leafy greens and fruit.
Here is some solid information that is a little more mainstream, and might be easier for your parents to swallow:
I have been vegan for over five years, and the only consistent supplement I have used is B12. I have taken calcium and plant based D off and on again too. I have never taken iron, never taken a multivitamin, and only for a short time in the beginning did I take vegan DHA. I've had my hemoglobin, B12, D, cholesterol, lipids, glucose, and other screening tests done over the years, a few this year, and all have been very healthy. I even restricted heavily for a year (2013) and still did not end up with any nutritional deficiencies, other than worsening bone density (I am also in surgical menopause and have numerous other risk factors that have nothing to do with being vegan).
I would recommend researching how to meet all your nutritional needs as a vegan so that you can be more confident in your decision and in explaining it to others. A good starting point is a book called "Becoming Vegan: The Complete Guide to Adopting a Healthy Plant-Based Diet" by Brenda Davis, RD. http://www.amazon.com/Becoming-Vegan.../dp/1570671036
The book is very comprehensive, uses science based studies, and does not cater to one style of vegan eating (ie raw, high carb, starch based, etc). It gives you basic information about macro and micronutrients and where to find them in a vegan diet. You can find this book in a lot of public libraries now, and you can get it used for very cheap on Amazon.com. This book was a lifeline for me in the beginning, and I still refer to it. Best wishes!