We investigated a mid-infrared (mid-IR) dual-band absorber consisting of a continuous gold film coated with an asymmetric silicon grating. In each unit cell of the grating, there are three unequally spaced silicon strips. Numerical results reveal that the (+1, -1) planar surface plasmon polariton (SPP) waves excited by the transverse-magnetic (TM) incidence can be coupled with different Fabry-Pérot (FP) resonances and the resonant energy is dissipated to the ohmic loss. Under the normal incidence condition, the absorber provides two high-absorbance peaks at wavelengths of 3.856 µm and 4.29 µm, with the absorption bandwidths of ∼25.7 cm-1 and ∼21.5 cm-1. When changing the angle of the incidence, it is observed an interesting feature that either of the peaks does not split. The presented structure offers an approach to the design of optical components for multi-spectral control of mid-IR signals.