The Carina Nebula Complex (CNC) is one of the most massive and active star forming regions in our Galaxy and contains several large young star clusters. The distances of the individual clusters and their physical connection was only poorly known up to now, with strongly discrepant results reported in the literature. We want to determine reliable distances of the young stellar clusters in the central Carina Nebula region (in particular Tr 14, 15, and 16) and the prominent clusters NGC 3324 and NGC 3293 in the north-western periphery of the CNC. We analyze the parallaxes in Gaia EDR3 for a comprehensive sample of 237 spectroscopically identified OB stars, as well as for 9562 X-ray selected young stars throughout the complex. We also perform an astrometric analysis to identify members of the young cluster vdBH 99, which is located in the foreground of the north-western part of the Carina Nebula. We find that the distances of the investigated clusters in the CNC are equal within <= 2%, and yield very consistent most likely mean distance values of 2.36^+0.05^_-0.05_kpc for the OB star sample and 2.34^+0.05^_-0.06_kpc for the sample of X-ray selected young stars. Our results show that the clusters in the CNC constitute a coherent star forming region, in particular that NGC 3324 and NGC 3293 at the north-western periphery are (within <=2%) at the same distance as the central Carina Nebula. For the foreground cluster vdBH 99 we find a mean distance of 441^+2^_-2_pc and an age of ~=60Myr, and quantify the contamination of X-ray selected samples of Carina Nebula stars by members of this foreground cluster.