You could start out by finding out how much demand there is for this kind of service. Go around your neighborhood and ask how likely everyone would be to utilize a dog walking or training business, and how much they would be willing to pay for such a service. Alternatively, you could create a survey online and send it to groups within your city. You can create a survey using Google Forms.
If there is enough demand for it and if the price is right, then present this information to your parents. This way you can show them that you're very serious about starting your own business and you have the know-how to do so.
Starting out with just a focus on dog-walking could be great, and then you could expand from there to training. If you have a good sized yard or a nearby dog park to work at for training, then you're set. After that you can move on to dog rescuing. Of course, you will need more space for rescuing dogs and will definitely need at least a yard.
Good luck with your parents and good luck with the business!