夸大其辞 and 夸大其词 are both attested and pronounced identically (kuādàqící). 词 (cí — 'word') is the more common modern form, while 辞 (cí — 'speech, statement') is the older variant. Both are correct.
history
Traceable to Song-dynasty scholar 楼钥 Lóu Yuè's 《攻媿集·论灾异》: '祥瑞之出，则润色而张大其词' ('When auspicious omens appear, people embellish and exaggerate them').
