While Mary didn’t ride a donkey, the magi did not have camels, and the centurion didn’t have a horse – though they all might have, but we just don’t know! – Jesus does ride a donkey in the gospels.
Mark (11:2-6) and Luke (19-29-40) both state that Jesus rode a colt on his entry into Jerusalem; that is, he rode an uncastrated male animal – a horse, pony, donkey, or mule – less than four years old. In this case, it is likely a donkey. In both cases, the colt has never been ridden on by anyone else. Jesus is its first passenger. John’s version is more abbreviated, simply noting that Jesus rode on a young donkey (John 12:14) with a quotation that is a fulfillment of not only Zech. 9:9 (as the synoptics allude to or cite) but also Zeph. 3:16. Matthew’s version (21:1-10) is a bit different: in it, Jesus sits astride two animals at once: a donkey and a colt. It is tempting to try to imagine how Jesus rides two animals at once, and this is sometimes thought to be either a careless or overly literal reading on Matthew’s part of Zechariah 9:9-10 (cf. Isaiah 62:11):
Rejoice greatly, O daughter Zion!
Shout aloud, O daughter Jerusalem!
Lo, your king comes to you;
Triumphant and victorious is he,
Humble and riding on a donkey
On a colt, the foal of a donkey. (Zechariah 9:9)
According to this view, Matthew misread the biblical parallelism – which the author Matthew cites as a composite citation of both Zechariah 9:9 and Isaiah 62:11 – as two animals rather than one. That is, Zion and Jerusalem parallel each other and stand in for each other, whereas donkey is paralleled and extended by colt, foal of a donkey. This overly literalized reading is a possibility. Also possible, and not really mutually exclusive with this explanation, is that Matthew regularly doubles material carried over from Mark; so, this may simply be another example of that.
What’s the upshot? Usually the Zechariah allusion and, in Matthew direct citation, is thought to represent humility (Zech. 9:9) and peace (Zech 9:10). It is a humble donkey and not a warhorse (cf. Ps Sol 17).
Finally, two more passages. The first we can dispense of quickly because we have already discussed it in the context of eating. In the middle of a parable of a great dinner, one of the excuses why people cannot attend is they just got some new oxen and want to try the out (Luke 14:19). Again, this is a work animal – though there are references to eating oxen – and its primary purpose was plowing; the oxen were the ancient tractor. Here it is a flourish of daily color that provides some verisimilitude. Matthew’s version does not have this excuse (Matt. 22:1-14).
Finally, there is a strange passage, which I am unsure what it ultimately means, that includes some working animals, again from Luke’s gospel account:
“Who among you would say to your slave who has just come in from plowing or tending sheep in the field, ‘Come here at once and take your place at the table’? Would you not rather say to him, ‘Prepare supper for me, put on your apron and serve me while I eat and drink; later you may eat and drink’?”
Luke 17:7-10
It is a dining scene that illustrates hierarchy: master and slave do not actually eat together. Not normally anyway. But it comes in the midst of slaves doing work: they are plowing the fields (again, this would involve oxen) or tending sheep for food and/or wool for cloth. In the broader context, Jesus is telling his disciples that they are in the position of slaves vis-à-vis God and do what they do out of obligation rather than for reward.