That's a deeply misleading summary of the data. The table excluding inflation is meaningless in the context of parent's comment, so I won't even touch that.
Real prices have dropped 14% in the past 19 years, and have been basically flat since 2004, except for an obviously anomalous dip during the financial crisis (2009/2010). The prevailing trend lines are a steady drop of 1.7% per year from 1995-2004 and then a flat-line (.15% increase per year) 2004-2014. To go back a bit further, airfare dropped over 50% 1978-2006[1]. That is, the trend in the late '90s didn't start in '95.
So no, parent's comment isn't quite literally true (prices have gone up 1.5% in the last decade), but historically prices clearly trend down, which I think is the point.
Airfares are most definitely higher than they were in the mid 2000's. The global recession saw a big cut back in capacity and that capacity hasn't been added back in to the system to offset the ever stronger economy. Flights are fuller than they've been in a long time meaning higher fares etc.
> Airfares are most definitely higher than they were in the mid 2000's
We've established that. The data shows that flights this year were about 13% higher than at the lowest point, which was 2009 (recession). They are a little under 5% higher than the lowest point during the "mid 2000's". These numbers (~.25% increase per year over 10 years) are fairly insignificant when you look at the broader trends (>1% decrease per year over 3-4 decades).
How much of that is down to taxes and fuel prices? Oil has dropped recently but for most of the past 5-6 years it's been much much higher than it was back then.
http://www.rita.dot.gov/bts/airfares/programs/economics_and_...
That shows 2014 has the highest CPI adjusted price since 2003 and the highest raw USD price going back as far as the data (1995).