The first thing that you need to do is to decide whether you want a permanent tattoo or a temporary one. It is strongly recommended that you go for a temporary tattoo, so that you can get it removed once you get bored of it. You will also have a much easier time convincing your parents to let you get a tattoo if it is just a temporary one and does not involves needles or anything of that sort.
Once you have decided what kind of tattoo you want to go for, figure out the design that you want. Keep in mind that it may only take 15-20 minutes to get a small-sized tattoo, whereas a larger tattoo can take from hours to days and even months, depending on the complexity of the design that you have decided to go for.
Visit a tattoo parlour in person, or use the Internet to find out how much it would cost to get it done. Compile the information that you collect, along with the design and size of the tattoo that you plan on getting, so that you can use it to make a case for yourself later. Make sure you include information about health and hygiene to your presentation.
Approach your parents and ask them for permission to get a tattoo. Hand them the presentation that you have prepared. Give them time to read through it. Do not become impatient.
If they give you the green signal, then that is great. However, if they say no, retreat for now.
Give it another shot after three-to-four days. If they say no again, retreat again. Continue to do this until they agree, or until you are no longer interested in getting a tattoo.