What most naturally follows?

How to convince your parents to let you become a dog walker
Take on more responsibilities.
Before approaching your parents, make sure to do your chores diligently for at least a week or two. Also, try to take on extra responsibilities in the house.
This way, you have material to use to prove to your parents that you are ready to take on the responsibility of having a job. For example, offer to help your younger siblings with homework or babysitting.