If the account is in your father's name, he may request a copy of the phone records.
This advise is rendered with only a brief description of facts which complicates any answer. You should use this answer to make the decision to contact a lawyer and explain your circumstances in detail for the purpose of getting accurate legal advise. The answer is based upon Arizona law.
No one other than the government is ever required to get a "warrant."
If the phone is in your father's name, he will have lawful access to the records.
I recommend you obey your father's rules until you are of age to make your own decisions and get your own cell phone.
The search warrant requirement applies to law enforcement agencies not parents of minor children who are checking up on their kids. In this case, the phone companies policies and procedures would mostly control what was required to obtain phone records.
Without reviewing all the facts and evidence, it is not possible to give a proper opinion regarding the liklihood of sucess of your case.
The other attorneys are correct. Search warrants are for law enforcement only. Phone records can be subpoenaed but if the phone is in your dad's name he can request the records directly from the phone company.