This paper proposes a new interval observer for continuous-time linear parameter-varying systems with an unmeasurable parameter vector subject to unknown but bounded disturbances. The parameter-varying matrices are assumed to be elementwise bounded. This observer is used to compute a so-called residual interval used for sensor fault detection by checking if zero is contained in the interval. To attenuate the effect of the system's uncertainties on the detectability of faults, additional weighting matrices and different upper and lower observer gains are introduced, providing more degrees of freedom than the classical interval observer strategies. In addition, a $L_{\infty}$ procedure is proposed to tune the value of the observer gains, this procedure being easy to modify to introduce additional constraints on the estimation algorithm. Simulations are run to show the efficiency of the proposed fault detection strategy.