They did. Both sides maintained Arab client kingdoms to fight each other, the Romans employed the Ghassanids in northwest Arabia and the Persians the Lakhmids in northeastern Arabia.

What was unique about the Arab conquests, however, was that for the first time Muhammad had been able to unite Arabia under what was effectively a state for the first time. Before Muhammad, no such organized unity had ever existed among the Arabs, who were divided regionally, tribally, and under various petty kingdoms (effectively the most organized of which were the Yemenis in the far south.)