Is it okay for my family to intervene in my career, workplace or professional life?
As a working professional, you and you alone are responsible for managing your interactions with your employer, manager(s) and colleagues. Your parents or any other family member, including spouses, have no role at all to play there. A parent who "intervenes" on behalf of their child, even an adolescent, will do irreparable harm to that person's professional reputation.
As a professional in the workplace, you are expected to manage your own career. Your parents' guidance will be invaluable, but they have no business discussing you or your performance with your employer. Family members should only contact the employer in case of emergency, when the employee is physically unable to alert his employer himself.
In a hypothetical situation where you are saddled with helicopter parents who somehow managed to contact your employer directly, the only response is to shut them down immediately to prevent it from happening again, and to apologise for their behaviour with your employer in a tone that suggests abject mortification. Anything less can make your manager think that you are unable to handle your own affairs and don't see your parents' involvement as the oddity that it is.
To quote Alison Green:
It's great for parents to coach their kids behind the scenes if the kids want it, but 20-somethings should be entering the workplace as the adults they are, which means interacting with their employers in the same way that other mature adults do.
Parents who get overly involved in their grown kids' professional lives and the employers who cater to them are performing a disservice, and are making it tougher for young workers to fully inhabit their new identities as independent, self-sufficient adults. They're denying them the opportunity to stand on their own, advocate for themselves, make their own mistakes and to be seen as competent, thoughtful, mature professionals.
Source: "Your Parents Don't Belong in Your Workplace", Alison Green on USNews.com, 2013-11-13
A quick note on those new to the workforce who don't yet qualify for "20-something" status: while you have a lot more leeway when it comes to professional norms, involving your parents in your professional life is still a Bad Thing. At the age when you're getting your first jobs, you are already approaching adulthood and as Alison says this is the perfect time to make mistakes. You will make them. But you'll make them on your own and learn from them. Having your parents hold your hand won't help you to learn anything and will come across as childish and immature. Part-time jobs and internships can be great references when you're looking for a "real" job and you don't want to be remembered as the guy whose parents came in to complain.
As some commenters pointed out, some jurisdictions may require the legal guardians to consent or sign off on any kind of contract involving a minor. Even then, you would typically have them read and sign the contract at home and return the signed copy to your new employer. While I can't think of any countries where they would negotiate a contract on the minor's behalf, if that is legally required and/or routine in your location, that would be the only exception I see to the "zero interference" rule.
This answer is harsh, perhaps unnecessarily so, but it's important that you (and people who stumble upon this question in the future) realise that it is simply not done to involve family members in your professional life like this. It's not that much of an issue in your specific situation and the workplace you applied at sounds dysfunctional at best, but it's an important lesson to learn. My advice? Thank your parents for intervening but ask them to let you handle situations like that in the future.