No, there is no space race to Mars.
The original space race was to orbit, for entirely military reasons. Sputnik was simply a demonstration of "we can drop a nuke anywhere we want, any time".
The second space race to the Moon was a publicity exercise. It served no military purpose, and it didn't serve any scientific purpose either. (Science builds off previous work. NASA deliberately destroyed all plans for the Atlas rockets, so that they could not be reproduced.) Primarily the reason was national pride (basically a willy-waving exercise with the USSR); and a strong secondary reason was a bread-and-circuses exercise in distraction from the war in Vietnam.
For Mars, we do not have either justification. A mission to Mars clearly isn't militarily useful. More importantly, the cost of the mission and the parlous state of the US and Russian economies mean that there isn't going to be any significant competition. China are actually in the strongest position nationally, since they have no political obstacles to redirecting funding at will. If there was an economic justification for going to Mars, then we could still get a space race from private investors trying to get a slice of the pie, but there is no economic reason.
Instead, what we have is a small collection of incredibly wealthy individuals funding private space research, with a fairly significant amount of government funding too. Inevitably someone will be first to get to Mars, but it isn't particularly a race, because all sides recognise that the important factor isn't getting there but surviving there. As private individuals without the artificial obstacles imposed by politics, this means we see a lot more collaboration. Space-X may have the best rockets, but it's likely someone else will have the best hab, for example.
Venus certainly hasn't been missed out on exploration. Previous posts have pointed out how harsh the environment is - you'd need to really want to have a rover there, and currently there's no good reason to do that. On Mars we have a good reason for wanting a close-up view of the surface, because we hope someday to send people there. Venus is interesting from a pure curiosity point of view, but the cost-benefit appears low when orbiters have already done high-quality mapping and observation work, and there is no prospect of a human ever walking on the surface. The airship concept is interesting, but it fails to explain why it is necessary to put humans on the airship, when the purpose is only to do more observation work which can easily be done remotely.