In order to reduce the possibility of structural fatigue and increase the lifetime of conveying fluid pipe, transverse vibration must be effectively eliminated. In this work, using parallel nonlinear energy sinks (NESs), a passive vibration controller is proposed to dissipate the vibration energy of the conveying fluid pipe. A high-order model of the conveying fluid pipe-parallel NESs system, in the form of partial differential equation, is derived and then converted into a quadratic form model containing the gradient information of a convex function. Combining the energy disturbance technique and first order convexity characteristic, the exponential stability of the closed-loop system is proved, which addresses the effectiveness of the proposed parallel NESs. Then, numerical simulations are given to verify the theoretical results and to illustrate the advantages of parallel NESs comparing with single NES. Finally, the reliability of the proposed approach is preliminarily verified through experiment.