Everyone likes independence in ones' life either it is kid,teenager or adult. Either it is in form of job or in the form of decision making power. In case of teenagers, if they do job with their study then I think they will know the importance of money and they will spend them carefully. There is a some reason below there which supports my statement.
To begin with, I believe teenager will gain self confidence if they do paid job with their study. They will learn money saving skills and will spend it where is the need. I remember, in my college days I used to do part time job. Because of this I didn't need to depend on anyone else. I used to do all of my work by my own.
In addition to that, I think it will make their financial condition better. Every parents want their kids gain good education but good education is very expensive and seems is not possible for everyone. Part time job will not only provide teenagers financially strong but also they will afford expencive education too.
Finally, They will gain work experience by doing paid job. Nowadays there is a very strong competition among graduates. Every year millions of talented students finishes their study and comes to the jobs. If you don't have work experience then it will become difficult to get the job. Experience will give them better future in jobs after their graduation. I remember in my college days I used to do making sites and it helped in my job interview.
By way of conclusion I am of the opinion that teenagers should do paid job while they are student it will not only make financly stron but also they will gain some working experience too.