Intrapartum fetal heart rate monitoring constitutes an important stake aiming at early acidosis detection. Measuring heart rate variability is often considered a powerful tool to assess the intrapartum health status of fetus and has been envisaged using various techniques. In the present contribution, scale invariance parameters, such as the Hurst exponent and the global regularity exponent, are estimated from wavelet coefficients of intrapartum fetal heart rate time series. Their ability to evaluate the health status of fetuses is quantified from a case study database, constituted at a French Academic Hospital in Lyon. Notably, the ability of such parameters to discriminate subjects incorrectly classified according to FIGO rules as abnormal is discussed. Also, the impact of the occurrence of decelerations identified as complicated by obstetricians on the values taken by Hurst parameter is investigated in detail.