Working part-time not only helped me make my own allowance but enabled me to appreciate the value of money.


Does "maky my own allowance" sound right? Thanks.
How about "earn my own living" instead? 
Annvan is correct. Your original sentence is unusual.
Do some searches before posting in order to see the patterns really used by other people. You should know how to do that by now.
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If this money is not what you use to live - to pay the rent, buy food, etc., then you can say "helped me earn my own spending money."

(Some people will tell you put the word "also" after but. Not only... but also...)
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As GG suggests, 'spending money' is a better term here. I consider 'allowance' to be something given by someone else. While I was growing up, my parents gave me a weekly allowance to spend (or save) as I chose.