You can have access to the Find Steed spell as a 2nd level paladin or a 13th level thief rogue. You would need to purchase or find the spell on a spell scroll.
The rules for spell scrolls state:
If the spell is on your class's spell list but of a higher level than
you can normally cast, you must make an ability check using your
spellcasting ability to determine whether you cast it successfully.
The DC equals 10 + the spell's level.
The paladin does not get spell casting until 2nd level, but may attempt to cast a higher level spell with the appropriate DC check. It's possible that the DM may rule that even though the paladin does not have spellcasting at level 1, the spell is on their class's spellcasting list, but this would be a per-table judgment.
At 13th level, the thief archetype of the rogue class gets the ability Use Magic Item, which states:
By 13th level, you have learned enough about the workings
of magic that you can improvise the use of items even when they are
not intended for you. You ignore all class, race, and level
requirements on the use of magic items.
Sage Advice has clarified that this ability does include spell scrolls:
Does the Thief’s Use Magic Device feature allow them to use spell scrolls?
Yes. The intent is that a Thief can use spell scrolls with Use Magic Device.
The downside of this method would be that if you ever dismissed the steed or let it fall to 0 hit points,you would need another spell scroll to cast it again.
13 levels of rogue is almost certainly too much of a dip, but 2 levels of paladin and a scroll might be justifiable. Also note that magic item acquisition can be up to the DM, so this might require discussion beforehand.