Paran is the area where Israel camped after leaving Sinai on their way to Canaan (Num.10:12; 12:16); the spies were sent into Canaan from there (Num.13:3,26); David went there after the death of Samuel (I Sam.25:1); and when Hadad the Edomite revolted against Solomon, he stopped there on his way to Egypt (I Kings 11:18). All of these clearly show that Paran could not be as far south as Mecca.
PARAN. A wilderness situated in the eastern central region of the Sinai peninsula, north-east from the traditional Sinai and south-south-east of Kadesh, with the Arabah and the Gulf of Aqabah as its eastern border. It was to this wilderness that Hagar and Ishmael went after their expulsion from Abraham's household (Genesis 21:21). It was crossed by the Israelites following their exodus from Egypt (Numbers 10:12; 12:16), and from here Moses despatched men to spy out the land of Canaan (Numbers 13:3, 26). The wilderness was also traversed by Hadad the Edomite on his flight to Egypt (1 Kings 11:18).
1 Samuel 25:1 records that David went to the wilderness of Paran on the death of the prophet Samuel, but in this instance we may read with the Greek 'wilderness of Maon'.
El-paran, mentioned in Genesis 14:6 as on the border of the wilderness, may have been an ancient name for Elath. Mount Paran of the Song of Moses (Deuteronomy 33:2) and of Habakkuk 3:3 was possibly a prominent peak in the mountain range on the western shore of the Gulf of Aqabah. (See also *ZIN.)
R. A. H. Gunner
("Baker Encyclopedia of Bible Places: Towns & Cities, Countries & States, Archaeology & Topography", Consulting Editor John J. Bimson, © Inter-Varsity Press, 1995, Published in the USA by Baker Books. ISBN 0-85110-657-9.)
Detailed discussions of the Muslim claims that Paran refers to Mecca are presented in these articles: , 
Answering Islam Home Page