If the parents owned property that was stolen (a laptop they gave to their son for school for example), then they can probably press charges.
Following the logic of the the post you quoted, the crime of murder could not be prosecuted in Tennessee because there would be no one with standing to press charges... It would be impossible to enforce our system of criminal law if victims were the only people who had the power to bring charges.
My understanding is that "Press Charges" is a lay term for filing a complaint. No citizen can file a criminal complaint in Tennessee. That's the job of the State through the District Attorney Generals' offices. Some States may allow citizens to press criminal charges, I don't know of any, but TN does not, as far as I know.
Any person can report a crime or bring a crime to the attention of the police, if they have personal knowledge of it. At that point, it is actually out of the hands of the victim and the case CAN proceed even over the victim's objection (although it
usually will not for various practical reasons).
The only thing a citizen should be signing with regards to an arrest warrant is an affidavit to support the warrant. Anyone with personal knowledge sufficient to attest to facts that create probable cause for an arrest can do that. Criminal law is replete with cases of informants signing an affidavit that resulted in a warrant for the arrest of someone.
Just a disclaimer: I'm not an attorney, this post shouldn't be confused with or taken as legal advice. I'm a poster on a message board and could be wrong about every last bit of this. Anyone who relies on this information does so at their own peril.