We are calling a stored procedure that accepts a very long comma delimited
parm. In the body of the stored proc, we loop through the parm using a user
defined function to pull out the tokens. The function call uses 2 input
parms: the original long parm and the int position of which token to extract.
When we run the code through the ODBC connection, the query dies after
calling the function 150-160 times, whereas when we run the EXACT same query
through Query Analyzer, it completes the necessary 340 calls to the function.
Is there some kind of stack limit or function call limit inside the ODBC that
QA is not experiencing?

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QA uses ODBC to connect to the Server, so I'm not sure what the issue is.
Perhaps you could turn on ODBC Trace when you execute the query using your
ODBC application, and then again using QA, and see if you see any
differences in the trace files that will solve your issue.