This paper presents an admission control mechanism for multi-rate wireless ad hoc networks. Admission control depends on precise estimates of bandwidth available in the network and the bandwidth required by a new flow. Estimating these parameters in wireless ad hoc networks is challenging due to the shared and open nature of the wireless channel. Available bandwidth can only be determined by also considering interference at neighboring nodes. Also, due to self-interference of flows the required bandwidth of a flow varies for each link of a route. The proposed admission control mechanisms is integrated with a hop-by-hop ad hoc routing protocol, thus enabling it to identify alternate routes if the shortest path is congested. Each node measures available channel bandwidth through passive monitoring of the channel. The mechanism improves estimation accuracy by using a formula that considers possible spatial reuse from parallel transmissions. The protocol also uses temporal accounting to enable bandwidth estimation across links using different bit-rates. Simulation results support that the admission control mechanism can effectively control the traffic load and that considering parallel transmission leads to improved bandwidth estimation accuracy. The admission control mechanism can admit more traffic while maintaining QoS.