Bonus HD from Dragon Rider probably do not cause your mount to advance age categories.
There aren't any errata to the Dragonlance Campaign Setting that I'm aware of, so there's no explicit rule that specifically says this for Dragon Rider mounts in particular. However, there are a couple of fairly strong reasons to think these bonus HD should not contribute to age category advancement.
Reason 1: Age categories are determined by age.
The True Dragon monster description tells you how dragons advance in age categories:
All true dragons gain more abilities and greater power as they age. (Other creatures that have the dragon type do not.) They range in length from several feet upon hatching to more than 100 feet after attaining the status of great wyrm. The size of a particular dragon varies according to age and variety.
Draconomicon describes this in more detail, starting on page 10. The section is too long to quote here, but it gives specific age thresholds at which dragons advance age categories.
The fact that age category advancement is clearly described as determined by age suggests that other ways of gaining HD do not result in advancing age categories. Being empowered by a skilled rider doesn't make your dragon older.
Reason 2: Bonus HD from class features don't grant HD-associated size increases.
True Dragons aren't the only creature that gets stuff at certain HD thresholds. Many monsters, such as the griffon, also increase in size category at certain numbers of HD, as described in their Advancement entries.
While the rules on this aren't as explicit as we might like, the consensus (e.g., Skip Williams, our own answers, Urpriest's monster handbook) is that bonus HD from class features, such as those gained by being a Druid's animal companion, do not grant such size increases.
Dragon age categories, as another secondary effect that normally occurs alongside HD advancement, should be expected to work similarly to these size increases.