We study small-amplitude axisymmetric shape oscillations of viscoelastic drops in a gas. The Jeffreys model is used as the rheological constitutive equation of the liquid, which represents a liquid with a frequency-dependent dynamic viscosity. The analysis of the time-dependent deformations caused by the oscillations yields the characteristic equation for the complex frequency, which describes the oscillation frequency and damping rate dependence on the viscous liquid behaviour and the stress relaxation and deformation retardation time scales ${\lambda }_{1} $ and ${\lambda }_{2} $ involved in the viscoelastic material law. The aim of this study is to quantify the influences of the two time scales on the oscillation behaviour of the drop and to propose an experimental method to determine one of the time scales by measuring damped oscillations of a drop. A proof-of-concept experiment is presented to show the potential and limitations of the method. Results show that values of ${\lambda }_{2} / {\lambda }_{1} $ from these measurements are orders of magnitude smaller than typical values used in simulations of viscoelastic flows.