It might of interest to you that cock-boat exists as proper word, "a type of small boat" (https://en.wiktionary.org/wiki/cock#Etymology_4). And it appears to be unrelated to the bird or the male member, although either etymology is not fully explained inasmuch as "child-talk" and "onomatopoetic" are short hand for the devil knows what they were thinking. Although it is obvious that the distant etymology plays hardly any role in the translation, it is likewise obvious that nobody can really say what your senior intends to say. I mean, it was not a vulgar insult challenging you to a cock fight.
So, in defense of your Dad and the English language at large, one may assume that the word is in use with wide ranging connotations. Describing these would have been your responsibility, or a matter for english.stackexchange.That's the rules.
You can't translate Blumenkohl nilly willy as '* flower coal': it's cauliflower. Hence I suggest it was wrong from previous answears to jump to conclusions. Cock - hahaha - may be a dirty word in some limited contexts, but that's not enough to translate cock-wagon as Penisprothese, "Schwanzersatz", "Schnäbli", "Schwanzverlängerung", "Potenzschlitten", even "Aufreißerwagen" (up-rip'er wagon, literally hook-up wagon) or anything like that, although Protzschlitten would be a good translation for vanity brands like Porsche.
Rather, the perspective suggests itself: youth would typically get small cars, initially, and maybe they'd cock them - is this applicable to cars, to arm up, "To lift the cock of a firearm or crossbow" [en.wiktionary]; as for German cf. auf-frisieren),instead of to tune, pimp, etc.? Either way the young men would be happier than ever and potentially (ha!) brag about it.
Incidently, Porsche is known for the Coupé (cf. Fench "cut up", "shortened").
protzig suits insofar it is akin to pretty, from Proto-Germanic *prattugaz (“boastful, sly, slick, deceitful, tricky, cunning”). See similar French petit, akin to petty (ha!).
More over, the word's form and collocation invites comparison to wheel and its reconstructed root *kʷekʷl-, akin to Ancient Greek κύκλος (cf. cycle); see also: Spanish chico "young boy", Latin ciccum "proverbially worthless object, trifle, bagatelle", Ancient Greek κίκκος; chickpea, Latin cicer "chickpea, (slang) testicle", Ancient Macedonian κίκερροι (kíkerrhoi, “chickpea”), etc. (supposed wanderword; cp. peanuts "little value", etc.). Although, this is not fully compatible with the boat sense, unless talking about a round boat, and a pathway for cock as borrowed from a comparable descendent from Proto-Indo-European *kʷékʷl-, which may also mean 'wagon' or 'chariot', seems to be unknown in the literature, I'd say it's a notable albeit unlikely avenue to pursue. Although *kʷekʷl- is the reduplication to *kʷel- "to turn", it would be necessary to assume here that the full grade was formed by analogy to the proper word.
In concusion, it's a matter of style. Translation practice offers enough freedome for a solution to interpret with a certain understatement
hübsche kleine Karre
nice little car
An all too literal translation instead would be Gockel + Karre (or Wagen, Schlitten, etc.), insofar as stolzer Gockel (proud cock) is a standing (ha!) collocation. But I doubt that this would be readily understood.
All this is pearls before the swines (Perlen vor die Säue) if the question was probably for one off use, for a homework in the worst case. If instead it was intended to settle an argument by reference to parallel constructions in other languages showing that it is vulgar, however, you may have to reconsider.