Ba(Ni_{1-x}Co_{x})_{2}As_{2} is a structural homologue of the pnictide high temperature superconductor, Ba(Fe_{1-x}Co_{x})_{2}As_{2}, in which the Fe atoms are replaced by Ni. Superconductivity is highly suppressed in this system, reaching a maximum T_{c}=2.3 K, compared to 24 K in its iron-based cousin, and the origin of this T_{c} suppression is not known. Using x-ray scattering, we show that Ba(Ni_{1-x}Co_{x})_{2}As_{2} exhibits a unidirectional charge density wave (CDW) at its triclinic phase transition. The CDW is incommensurate, exhibits a sizable lattice distortion, and is accompanied by the appearance of α Fermi surface pockets in photoemission [B. Zhou et al., Phys. Rev. B 83, 035110 (2011)PRBMDO1098-012110.1103/PhysRevB.83.035110], suggesting it forms by an unconventional mechanism. Co doping suppresses the CDW, paralleling the behavior of antiferromagnetism in iron-based superconductors. Our study demonstrates that pnictide superconductors can exhibit competing CDW order, which may be the origin of T_{c} suppression in this system.