Is the parameter of interest uniformly distributed over the confidence interval? Let's assume that we are estimating a parameter of the population (e.g., mean height or weight). We've gathered our sample and calculated confidence intervals around the parameter of interest. Are we, therefore, assuming that this parameter follows uniform distribution over the confidence interval? In simple terms, is the parameter equally probable to be anywhere within this confidence interval?

If you are referring to frequentist statistics, in that case, when you repeat an experiment, and construct the 68% confidence level confidence interval each time, then 68% of those intervals should contain the true value.  The true parameter is fixed, and does not follow a distribution - the confidence intervals do.
If you are referring to Bayesian statistics and credible intervals, then the posterior distribution of the parameter depends on the prior and on the likelihood.
