I like the feel of this, especially the nostalgic thoughts combined with a present day moment. There is some really nice imagery but I feel it needs just a bit of work in the following way:

and equine limbs that filled the air
and nostrils flared;
all animal, all heart and blood;
all scared and furious and free.

I think with all the 'ands' (5), 'with' would work much better there. I personally think you could lose the first one and it would still work very nicely.

I'd rather drive a horse
and then we'd have that;
something beating and breathing and true,
not a metal, man machine with wheels and valves -

Again with 4 'ands' I think losing the first would strengthen the repetition of the remainder.

we know exactly what to do and how

I think knowing exactly what to do implies and how for the most part and is a much stronger ending.

to catch a horse
(to calm a horse -
a thing that's never felt the metal.
Hooves are like fingernails, they don't feel.)
it doesn't take Super-Dad courage

If you lose the parenthetical reference, you wouldn't need the "it". Also, the parenthetical reference feels a bit too disjointed and isn't flowing well.

I never thought that all you were for

In addition to losing 'that, the voice throughout this is a bit confusing. I'm uncertain who you're addressing. You start out in a car enroute to the theatre and you're reminded of a memory. But I'm uncertain who the "we" is.

Then you refer to your dad in the next verse. The third verse is referencing the evening before your dad caught the horse, then in the next stanza the voice completely changes to: I stopped letting "you". So the "you" addressed in this voice change is the same part of "we" in the first stanza? There's something missing here, a smoother transition and some clarity, but I can't quite put my finger on it.

Overall I enjoyed the nostalgic memory of your father and the horse metaphor in reference to a relationship. I just feel the voice clarity and some transitions need a bit of work.

Well done.