Network flow fingerprinting can be used to de-anonymize communications on anonymity systems such as Tor by linking the ingress and egress segments of anonymized connections. Assume Alice and Bob have access to the input and the output links of an anonymous network, respectively, and they wish to collaboratively reveal the connections between the input and the output links without being detected by Willie who protects the network. Alice generates a codebook where each codeword is a unique fingerprint indicating a sequence of interpacket delays, and shares it only with Bob. To trace each flow, Alice selects a fingerprint and manipulates the packet timings of the flow to follow the packet timings suggested by the fingerprint, and Bob extracts the fingerprints from it after it passes through the network. We model the network as parallel M/M/1 queues where each queue is shared by a flow f_i from Alice to Bob and other flows independent of f_i. Packet timings of the flows are governed by independent Poisson processes. Assuming all input flows have equal packet rates and that Bob observes only flows with fingerprints, we first present two scenarios: 1) Alice fingerprints all the flows and 2) Alice fingerprints a subset of the flows, unknown to Willie. Then, we extend the construction and analysis to the case of arbitrary flow rates and the case where Bob observes flows with and without fingerprints. For each scenario, we derive the number of flows that Alice and Bob can trace by fingerprinting.