You cannot deduce anything from the data. You may conclude, but not all conclusions are deductions. To conclude that there is or is not a contrast in initial position, you have to know the rules of the game. A possible (common) "rule of the game" is "if there are minimal pairs distinguishing X and Y in a position, then X and Y contrast in that position". There are no minimal pairs in initial position in these. However, the minimal pair test is not the only rule of the game. An alternative rule of the game is based on whether there is a rule that derives X from Y in that context. In that case, you would also have to figure out what the rule would be deriving X from Y.
Another rule-of-the-game problem (a veritable universal problem in the profession) is that if you have proof that two things contrast, you can say "yes", but you can never say "no", because the question isn't stated as being dataset-relative (i.e. "in these examples..."). For all we know, the next word will present us with a minimal pair. If we assume that "no" is possible for the case where there is no evidence, then "no" would be correct.
You should be proud of yourself for understanding the concept "phoneme" well enough to see that even though there are contexts where the contrast is neutralized, those sounds are distinct phonemes. I disapprove of the metaphor that using X vs Y "changes the meaning": the word "dog" didn't start out meaning "but" and change (nor the converse), instead one phoneme occurs in one word (peɾo), and a different phoneme appears in another word (pero). Incidentally, "dog" is [pero], not [per̃o], which indicates a nasalized r in IPA. The evidence that the data are in IPA is pretty much non-existent – an example like rancho would clarify what system is being used.
Elaborating on the alternative to requiring minimal pairs ("can you write a rule predicting X from Y"), here is a hypothetical dataset (because it's made up and I assert that these are different words, we don't need glosses). The question is whether [t] and [r] contrast.
tima, tusa, tope, tada, temu, tumi, tiso, rino, raka, ropo, ramu, rapu, rodo, raga
There are no minimal pairs: does that prove that these are separate phonemes? It does not, if you can provide rules that derive all cases of [t] from /r/ or vice versa. It may seem like deriving one from the other is impossible. However, [r] mostly appears before [a] and [o], never before [u] etc. More than one rule is necessary: t→r/_a plus non-coronal consonant; t→r/_in; t→r/_oCo. Given these rules, r can be treated as an allophone of /t/. A standard objection is "But that requires a lot of rules", and the proper response to that is "So how does that invalidate my analysis?". There is another rule of the game, to the effect that if you possibly can treat X as an allophone of Y, you should. I think that rule is bad, and you need something more than simply "because I can eliminate that phoneme" as a justification.