So long as you only target yourself you can take the mount.
The crux of the question is whether Dimension Door qualifies as a spell that "targets only you" as required by Find Steed.
There is an argument that Dimension Door targets a point in space and so would not qualify. I suggest, however, that interpreting the "spot" mentioned in the spell as a target does not match how targets are discussed in the rules.
The rules for targeting mention targeting a point in space only in reference to the point of origin for an Area of Effect:
For a spell like fireball, the target is the point in space where the ball of fire erupts.
A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area of effect
There is no indication that a space can be a target except as the point of origin for an area of effect. The second quote indicates that there are three types of targets: creatures, objects, and points of origins for AOEs.
Unfortunately the description of Dimension Door makes no explicit mention of "targets" but we can infer from the text that you and whichever creatures you take with you qualify as targeted creatures, fitting the first category of "target". Furthermore the spell does not describe the point of origin for an area of effect.
If the Bard wanted to take another person with them when using Dimension Door then they would be unable to take their mount because the spell wouldn't target only them.
If, however, the bard wanted to teleport just themself, without taking an additional creature as allowed by the spell, then the rules for Find Steed would allow them to also bring the mount.
If a DM thinks that the space is also being targeted and so disallows Dimension Door you could always take the Mastiff, a medium mount instead considering that RAW it can still serve as your mount.