We report a \(^{139}\) La-NMR study of \(\textrm{Ba}_{2}\textrm{La}_{2}\textrm{CoTe}_{2}\textrm{O}_{12}\) , \(S=1/2\) equilateral triangular-lattice antiferromagnet with easy-plane anisotropy at low temperatures. This compound undergoes a magnetic phase transition at \(T_\textrm{N} =\) 3.26 K into an ordered state with the \({120}^\circ\) spin structure. Under magnetic fields above 3T, \(T_\textrm{N}\) splits into \(T_\textrm{N1}\) and \(T_\textrm{N2}\) , which correspond to the transitions from the paramagnetic phase to the up-up-down (uud) phase and from the uud phase to the triangular coplanar phase, respectively. The NMR spin-lattice relaxation rate \(1/T_1\) exhibits a critical divergence at \(T_\textrm{N1}\) , indicating the onset of long-range magnetic order. At \(T_\textrm{N2}\) , the NMR-linewidth measured at 5.4 T exhibits an anomalous decrease, which we attribute to a change in the spin structure from the uud to the triangular coplanar phase.