Here is an article on shoulder dystocia research (it's pretty long).
This is about the multiparity issue:
In a 10-year series collected from Boston's Beth Israel Hospital covering the years 1975 to 1985, Acker (1988) showed that there were more Erb palsies in babies born to multiparous women then to primigravida women. He attributed this to a marked increase in precipitous labors in such women. In his series he noted that 31.8% of all babies with Erb palsy had experienced a precipitous delivery.
But as with maternal age, by the time a woman becomes "multiparous", she is old enough to have an increased risk of having other risk factors for shoulder dystocia such as larger babies, obesity, and diabetes. Moreover, only multiparous women could have the very significant risk factor of having had a previous shoulder dystocia. Thus most experts feel any relationship between multiparity and shoulder dystocia is secondary to other, more primary, risk factors.