|"Red" Rudy Williams|
Who's playing the alto saxophone on this one?
The musician playing alto on the 1948 track you heard which was recorded under Tadd Dameron's name is "Red" Rudy Williams, a musician Charlie Parker used to dig intensely, almost religiously, during his first visit to New York City.
Member of a hard swinging band, Al Cooper's Savoy Sultan, he was usually featured on radio broadcasts some ten years before the above recording was waxed. (Other musicians featured on the piece you heard are Fats Navarro, Allen Eager, Curly Russell, and Kenny Clarke.)
In 1939, Williams was enjoying the success of this hit song, Little Sally Water, in which his name is called by Savoy Sultan's before he moves to front for a solo:
A fellow musician Biddy Fleet with whom Parker spent a considerable amount of time practicing and
exploring new possibilities in jazz remembers:
"On any give night, Charlie might ignore the radio until the Sultans came on. As Williams's alto sound surged out of the speaker, he would exclaim, 'There's Red Rudy!'"
Looney by Al Cooper's Savoy Sultans from July 29, 1938
Fleet, observing such enthusiasm for a player who considered inferior to Parker, was surprised:
"Williams was a straightforward player, nowhere near as adventures as Charlie was, or wanted to be."
Fleet argues that Parker's interest in Williams might have something to do with the cold response he received during his first New York jam sessions. Fleet interprets it as Bird's nod to compromise or his contemplation on softening his sound.
"Sometimes I wonder if I should play like Rudy Williams," Charlie mused to Biddy Fleet. 
In retrospect, it seems if Parker stood firm and didn't change directions Rudy's way, it was Rudy who decided to emulate Parker. Listen to this for instance (or go back and listen one more time to the opening track by Dameron)
That was Savoy Jam Party (Part I) by Don Byas Orchestra, featuring Charlie Shavers (tp) Rudy Williams (as) Don Byas (ts) Clyde Hart (p) Slam Stewart (b) Jack "The Bear" Parker (d), recorded in New York, August 17, 1944.
Al Haig, before acquainting Parker and becoming one of his favorite pianists, confirms the notion of hunter being hunted by the game:
"Rudy [with whom I had worked in the Savoy] had heard Charlie Parker. He knew what he was doing [when] I didn't even have an idea who Charlie Parker was." 
There is more Rudy Williams to be heard on recordings by Eddie "Lockjaw" Davis, Gene Ammons, Billy Kyle, Illinois Jacquet, and Howard McGhee.
 All quotations, except the last one, from Kansas City Lightning by Stanley Crouch.
 Swing to Bop by Ira Gitler.