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Can I start driving alone as soon as I got my driver's license?

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I just took the road test today which I passed, and so I received a temporary sheet for the license while I have to wait a week from now for the actual DL card to be sent to me. As I call today as my first day with the DL, can I can start driving alone already? Somehow, my parents told me I have to wait because my name is not recorded on the insurance yet and because in that case without my name being on the insurance, I would be held responsible or whatever in a stricter consequences if I ever get in a car accident.. and I live in Texas in case if you happen to know more about the laws in TX.