Age of the universe and time dilation
The answer is yes time dilation does affect how much time an observer experiences since the big bang until the present (cosmological) time.

However there is a certain set of special observers called comoving observers, these are the observers to which the Universe appears isotropic to. For example we can tell the Earth is moving at about 350 km/s relative to a nearby comoving observer by measuring the anistropy in the cosmic microwave background (in fact it causes a relatively large anistropy, the pictures of the CMB you see from WMAP, etc are with this anistropy factored out).

One special property of a comoving observer in the Universe is that they maximize the age of the Universe, i.e. no other observer can experience more time since the big bang than a comoving observer (in case you ask there is no observer that minimizes the time since the big bang). When we talk about the age of the Universe we are talking about the age of the Universe from the pov of a comoving observer.