You Want A Syrinx Instead of A Larynx
Parrots can modulate their voices and produce a far wider range of sound because instead of a one-way Larynx they posses a Syrinx. With humans our voices are produced by the air running through a single passageway in our throat. A Birds Syrinx is an inverted Y shape with an expandable air sac that allows air from both lungs to be used at the same time like ours, or only air from one lung at a time. Additionally instead of vocal chords they have a vibrating membrane attached to this air sac. By expanding and contracting this air sac they can not only more accurately control the airflow across this membrane than we can, but also change the resonance of their voices as well. Something one can only kind of do with expandable/contractible vocal chords within a larynx.
This means that a person who had been somehow bio engineered to posses a functioning Syrinx could cover a far wider range of sounds and frequencies than a normal human. Here's a highly simplified picture of how a Syrinx works.
Keep in mind that when a person is caught by police who happens to have exactly the wrong body part needed to say that they were capable of making the voice in the phone call then they are as good as caught anyways. Unless having a Syrinx installed is all the rage and everyone has them then if your person gets connected to the call by other evidence its as good as a smoking gun that they made the call. Kind of like how burning your finger-prints off would make you pretty much the only guy who leaves blank fingerprints at a crime scene and thus wouldn't make you any harder to connect to the crime or any less reliant on wearing gloves to avoid leaving prints.