In this paper, we present various optical metamaterial nanoabsorbers with the aim of improving the refractive index sensitivity using the Fano response. The proposed absorbers consist of various parasitic elements such as single cross, broken cross, Jerusalem cross, and also single L and double L models. We numerically study their absorption and reflection using the three-dimensional finite-difference time-domain method and calculate the sensitivity and figure of merit (FOM) in every absorption peak (reflection dip). Our simulations reveal that the Fano resonance at longer wavelengths can be used for increasing the sensitivity and FOM. The proposed absorbers have been coated with an external material with a maximum thickness of 100 nm and refractive indices in the range of 1.05–1.2. We also study and compare the FOM for these structures. They are modified for 900 and 1200–1300 nm. The maximum FOM is achieved around $2400{\mathrm{RIU}}^{-1}$ for the double L nanoabsorber coated with 1.05 indexed material, while its sensitivity is 473 nm/RIU. This absorber is an appropriate component for the design of highly sensitive optical sensors.