I can certainly understand why your parents are concerned about their 16 year old daughter flying halfway around the world to a place that has had, shall we say, some unfortunate incidents and publicity during the past several months. I was there during the election and subsequent riots. I was not discouraged, but I'm an adult with a lot of international travel experience. What is the heart of your parents fears? The flights? Umm, they do know air travel is exceedingly safe, don't they? The airports? Transiting at Schipol or Heathrow? Both of those airports have websites with maps and all sorts of info. Perhaps looking at them will allay their concerns. I prefer to change planes at Schipol over Heathrow (that's just personal - I've had a couple of near misses traipsing across Heathrow to my connecting flight). Schipol is a manageable size, there's a food court, youcan even point out the mini branch of the Rijksmusem. Everybody (people working at the airport you will come in contact with) speaks English there and you can pay for snacks etc with plastic. Landing in Nairobi can be chaotic. It's not the landing, or going through immigration, or picking up your luggage, but when you walk out of the terminal there will be numerous touts and taxi drivers. I don't think twice about ignoring these people, but it may be a shock to you if it is something you haven't encountered. It is best to have transport arranged, either with your friend or by a hotel that will have taxi driver holding your name up on a sign. You will also likely be disoriented due to the length of the combined flights, being tired, the time difference, and it will likely be nighttime when you arrive.
Are your parents concerned about your friend? Is this someone you met online? If so then your parents are justifiably concerned. Have your parents met this person? What is the nature of this volunteer work? Is it through a school, church, or other charity? Will you have access to resources, other chaperones, etc if staying with your friend doesn't work out? That is an issue you should be able to address for your parents. What will be the process if you have a medical emergency?
Also, you may want to check if you need any additional documentation, i.e. notarized letters from your parents authorizing you to travel abroad, as you are a minor.
I am not trying to be negative. I think going to Kenya is an excellent idea (I was just there) and I would have killed to do so when I was 16. But I can certainly understand why your parents would be concerned (especially if they've never been there) and think your best chance of getting them to agree to your going (and probably paying for it) is if you have a plan that can allay their concerns. Good luck.