How probable is the best fit of a model to lie outside 68% limit constraints on model parameters? I am running MCMC analysis on a model using an observation data point. After the MCMC run has converged, when I check the best fit parameters and compare it to the constraints of the model parameters, I find that they are sometimes within 99% confidence limits of the model parameters but not 68% limits. How probable, is it? or am I making a mistake somewhere?

I think this can happen if your posterior distribution is highly skewed or otherwise non-normal. What you haven't said is how you are deciding what the "best-fit" parameters are or which 68% condfidence limits are being quoted.
If the best-fit is the median of the PDF then I don't see how this can happen. However if you choose the peak of the PDF whilst the confidence limits are taken from the mean or median of the pdf and where the $\pm$ 34% of the integrated PDF is from that, then it can happen.
An example would be where say your prior on the parameter runs from zero to infinity. Maybe the peak of the PDF is very close to zero - or in fact zero - and the PDF declines monotonically towards higher values. The "best fit" would be close to zero, but you confidence limits might be quoted as lying $\pm 34$% from where the median of the PDF is.  (Dodgy sketch below).

