Paid time off is not a legal right employees are entitled to. It is a benefit employers may choose to grant to employees. When offered, the employer may place any terms and conditions on how such a benefit may be implemented. The only limitation is that once the benefit is earned, it cannot be forfeited.
However, there is nothing unlawful about an employer requiring approval in advance of using this benefit. If they do not wish to allow two employees off at the same time, that is within the employer's discretion.
There may be some exceptions to the applicability of a blanket policy, for example, if you were requesting an accommodation for a religious holiday or medical condition or some other protected need for time off, the employer would have to attempt to accommodate the request. But employees do not have the right to take time off whenever they want to, simply because they have PTP.
They say you get what you pay for, and this response is free, so take it for what it is worth. This is my opinion based on very limited information. My opinion should not be taken as legal advice. For true advice, we would require a confidential consultation where I would ask you questions and get your complete story. This is a public forum, so remember, nothing here is confidential. Nor am I your attorney. I do not know who you are and you have not hired me to provide any legal service. To do so would require us to meet and sign written retainer agreement. My responses are intended for general information only.
The previous attorney who addressed your question is correct. Employers are under no legal obligation to provide PTO--it's a job perk, and as such, can be regulated in any fashion the employer chooses. The only notable limitation is that accrued PTO generally cannot be forfeited. Employees must eventually be allowed to take it or, upon separation of employment, be cashed out for their remaining balance.
This answer is a general interpretation of the law and is not fact specific to your case. Likewise it does not create an attorney-client relationship. You should seek an attorney for a review of your specific facts and documents.
California does not require an employer to provide paid vacation. As a result, the employer may set the terms and conditions for taking PTO, including but not limited requiring advance notice and not allowing multiple employees to take PTO at the same time. Please note the employer may not take away your accrued PTO and must cash it out when you separate from your employment.
As a practical matter, you may wish to speak with your co-workers to try and coordinate your vacations so that they do not overlap.