"From the start, one of Charlie's goals had been to refine his tone, to get rid of that goddamn vibrato and create a sound that was built for speed. To further his cause, he took a Brilhart mouthpiece---an uncommon brand at the time---and experimented with its shape, filing it down in an effort to control the sound even further. In doing so, he was ignoring the advice of older musicians, who warned him about the possibility of brass poisoning. Players such as Tab Smith, an alto player with Lucky Millinder and Basie, had gotten infections when the sharp edge of an altered metal mouthpiece cut into their lips as the played. As was his way, Parker listened, said nothing, and then went on with what he was doing. Parker became such a proficient reworker of mouthpieces that he was soon filing away on those of his fellow band members and of any other saxophone players who trusted his skill."While this is the first I have heard of Bird working on mouthpieces, it certainly is not unthinkable that he did. From Earl Bostic (with his [apocryphal?] razor-blade baffle) to Jerry Bergonzi (who is known to have adjusted his own mouthpieces) to a contemporary young player like Matt Marantz (who has paying customers for his mouthpiece modifications), jazz saxophonists have long engaged in re-crafting their mouthpieces.
There are however a few implausible elements to Crouch's claim. First, the Brilhart mouthpiece that Bird played was made from "Tonalin," a kind of plastic, so he would not have been at any risk of "brass poisoning." Bird does not seem to have played a metal mouthpiece on a regular basis until 1949. Secondly, while the Brilhart "Streamline" mouthpiece that Bird played was first manufactured in 1939, there is no evidence to suggest he played it until 1945, after he had left Kansas City. Instead, as saxophone technician Nicholas Trefeil documents, when Bird was with Jay McShann around 1940, he was playing an Otto Link "ResoChamber."
The additional matter of precisely what it means when Crouch says that Bird "fil[ed his mouthpiece] down" and whether this is the best way to put it I will leave to the mouthpiece technicians.
In any case, it is well known from photographic evidence that Bird played many different mouthpieces, and it is not at all unlikely that he would have attempted to adjust them himself in pursuit of his idiosyncratic sound. A sound which was, after all, perfect.