You have your query wrong, instead of:
UPDATE parts WHERE PartID = $partID SET $updateData;
It should be (note the order of things):
UPDATE parts SET column_name = 'column_data' WHERE PartID = $partID;
Take a look at UPDATE Syntax for more information.
About the error :
mysqli_error() expects parameter 1 to be mysqli
$mysqli is the variable that holds your mysqli, that's the one that should be used:
NOTE: I don't know where this data is coming from, but in case it's user input, you should definitely take a look at How can I prevent SQL-injection in PHP?