How is this conflict about age of the universe resolved? In a previous Phys.SE question, Does a spaceship travelling at near lightspeed see the universe aging slow or fast?, the answer (which was followed by a proof involving co-moving reference frames) was given as 

The short answer is that yes, an astronaut moving relative to the cosmic microwave background would measure a shorter time since the Big Bang than an observer stationary wrt to the CMB. 

However, an observer in such a spaceship will consider the time of any object which is at the CMBR co-moving reference frame to be moving slower than itself.  Is this not a conflicting result?
For example, let's say the spaceship and a CMBR Earth communicate as they pass by each other.  Each would have an estimate of the age of the universe, and each would have an estimate of the measured age of universe that the other would give, based on their own measurement of the age and the time dilation that they assume the other would experience.  Here are the results

The CMBR observer is fine - both his estimate of the universe's age that the spaceship would give and the estimate actually given by the spaceship match and are less than his own estimate of the universe's age.  However, the spaceship expects the CMBR observer to have a lower estimate of the age of the universe because their clock is (according to spaceship observer) ticking slower than his own.  What the spaceship observer does not expect is that the CMBR observer's estimate is larger than his own estimate of the age of the universe, yet that is what happens.  How is this resolved without implying a preferred reference frame?    
 A: 
How is this resolved without implying a preferred reference frame?

That you asked this implies you are thinking that reference frames have universal extent. While that is true in special relativity, it is not the case in general relativity. Reference frames are local in general relativity.
That said, there is a frame in which cosmologists prefer to work, and that is a frame locally at rest with the cosmic microwave background. This is the frame that yields the longest proper time for the light from the surface of last scattering. That does not mean that this is the "preferred frame" (with all the baggage that goes along with that term). It is just the frame in which cosmologists prefer to work.
A: While the spaceship accelerates the time on earth is also running faster in the reference frame of the spaceship. The fact that everybody sees the other's clock tick slower is only true for uniform motions. The time dilation coming from gravity and acceleration, which are equivalent, is absolute. So if the spaceship does the calculation right it can easily transform into the system which is at rest to the CMB and find out the age of the universe in that frame of reference.
If the spaceship is created at rest (relative to the CMB) it will have to accelerate to gain velocity in the direction of the earth. Even if the spaceship is created at motion from the beginning there is the so called "relativity of simultaneity".
