The Dragon Disciple's casting requirement is the "ability to cast 1st-level arcane spells without preparation." This is clearly intended to limit the DD to being entered by Sorcerers without straight up limiting it to Sorcerers.
Preferred Spell lets my Prepared Caster spontaneously cast one spell as many times as I have spell slots of high enough level.
Now my question is this. Do I need to be able to cast two or more different spells without preparation (meaning take Preferred Spell twice), or does simply being able to cast my Preferred Spell more than once count as "spells"?
I believe there's actually a difference between 'cast spells without preparation' and 'spontaneous casting'.
The only references in the core rulebook to 'spontaneous casting' are in reference to the Druid and the Cleric, both who get the ability to spontaneously cast certain spells in much the same way that Preferred Spell works. I mean, the section called 'Spontaneous Casting and Metamagic Feats' is entirely about clerics and druids. Bards and sorcerers get their own section above.
This is muddied a bit by the Prestige Classes referring to 'spontaneous casters' in their Spells per Day section, so perhaps it's just an academic distinction. But it's an amusing one none the less.
Bards and Summoners also qualify for DD, not just Sorcerers.