What's the oldest possible age of the universe? (LCDM parameter range) What would happen if you took today's six-parameter LCDM model down to the 5% confidence level?  Understanding that the number of permutations is high, I would be particularly interested just in the present age of the universe t0---- what are t0's higher and lower limits, assuming that all five of the other parameters are leaning all the way to the plausible floor at the 5% CL either to increase t0 together, or to decrease t0 together?
Q: What would happen if you took today's six-parameter LCDM model to the 95% confidence level? Understanding that the number of permutations is high ...

NASA's Graphical Parameter Comparisons webpage has this to say:

Sources in the literature for values of these selected parameters determined between the years 1996 and 2016 are shown in the Data References Table. It would be a formidable task to include all results from all experiments and every paper. Therefore, results presented here are representative, and there are multiple caveats associated with those choices. For example, we do not guarantee that the datapoints are statistically independent of each other, although the choice of data sources is motivated to some extent by a wish to include independent determinations. There also has been no attempt to estimate what fraction of the parameter uncertainties are due to cosmic variance, although this should clearly be a consideration.
We discuss the individual parameters in pages that follow, and include an accompanying graphical history for each. Vertical gray lines shown in the graphical histories are weighted averages of the WMAP and Planck data points rather than the entire sample. This is done as a visual aid, rather than a statement that this is the current best value: the WMAP and Planck data values serve as a common thread in each of the plots. In a few cases, liberties were taken in the quoting of asymmetric error bars from published results as symmetric error bars on the plots. For illustrative purposes, the differences are negligible. Quoted uncertainties are 68% confidence limits.
Data references Table - Contributed by the NASA LAMBDA Archive Team.

The published data isn't accurate enough to reach the confidence level you require.
Prof. Wright of UCLA has a webpage about the age of the universe using actual age measurements, not estimates from cosmological models, but a more authoritative source would be NASA's Lambda Archive Team which has a page on $t_0$.
There are many ways to tweak the values and make the calculation, the computed value is somewhat dependent of the exact cosmological model assumptions. Within the standard ΛCDM model framework, CMB measurements, combined with BAO and SNIa measurements of the expansion rate, are presently in agreement and tightly constrained. The gray vertical line, representing the weighted average of WMAP and Planck data points, is positioned at $t_0$ = 13.796 Gyr.
This graphic from their website sums it up: