The Bayesian conception of a probability is not necessarily subjective (c.f. Jaynes). The important distinction here is that the Bayesian attempts to determine his/her state of knowledge regarding the value of the parameter by combining a prior distribution for its plausible value with the likelihood which summarises the information contained in some observations. Hence, as a Bayesian, I'd say that I am happy with the idea that the parameter has a true value, which is not known exactly, and the purpose of a posterior distribution is to summarise what I do know about its plausible values, based on my prior assumptions and the observations.
Now, when I make a model, the model is not reality. So in some cases the parameter in question does exist in reality (e.g. the average weight of a wombat) and in some questions it doesn't (e.g. the true value of a regression parameter - the regression model is only a model of the outcome of the physical laws that govern the system, which may not actually be captured fully by the regression model). So to say that there is one true fixed parameter value in the real world is not necessarily true.
On the flip side, I would suggest that most frequentists would say there is one true value for the statistic, but they don't know what it is either, but they have estimators for it and confidence intervals on their estimates which (in a sense) quantifies their uncertainty regarding the plausibility of different values (but the frequentist conception of a probability prevent them from expressing this as directly).