> In Texas, minors aged 14 and older can work without parental consent. If an employer chooses to do so, he or she may require the youth to submit a “Certificate of Age” form.

(n.b. 2017)

https://texaslodging.com/child-labor-laws-2017/

> Now minors [in New Jersey] will no longer need parental consent or anything from the school district to secure a work permit. Instead, the state Labor Department will create a centralized database for teens to register for work permits, which will be accessible to employers when interviewing potential employees.

https://newjerseymonitor.com/2022/07/06/teens-in-n-j-can-wor...

Instead of posting more links I will just summarize what I am finding... Instead of parental consent most states require a work permit which is more formal documentation than just a parent saying ok. Obtaining a work permit includes a review process, and require the submission of various materials which sometimes includes written permission from a parent, for some age groups.