Is syllable a phonetic or a phonological concept? Consider 'syllable counting' as a task: would that be regarded as a phonetic task or phonological tasks? Would it depend on whether words are presented visually or auditorily?
A syllable is a phonological unit. Native speakers of two different languages can hear the same stimulus as having different numbers of syllables if their respective phonologies have different criteria for what can constitute a syllable.
For example, a Hebrew speaker may hear a stream speech and perceive it as [pka] while an English speaker might hear the same stream of speech and perceive it as [pə'ka], since Hebrew phonotactics allow syllable onset kt-clusters while English phonotactics don't.
What constitutes a syllable in a given language is deduced by linguists using phonological information gleaned from various criteria, such as stress/accent/tone assignment, word games, metrical divisions in poetry and music, and orthography (in cases where a language's orthography is syllable-based).
Syllable counting is a phonological task that relies on learned knowledge about the phonotactics of the language in question. It requires words to be presented auditorily or in some unambiguous phonetic transcription, since orthography is often not a reliable or unambiguous representation of pronunciation (How many syllables in the English word peer? How about 911?).
Phonetics is also relevant, of course, since interpreting a stream of speech and mapping it to its phonological representation involves knowledge/awareness of the phonetics of the language at hand. In English, for example, the location of a syllable boundary in relation to an [st] cluster is cued in part by how long the [t] closure is and how long the period of aspiration before the vowel is. In [as.ta] the [t] will have a longer closure and a longer period of aspiration than the [t] in [a.sta].
One last note: even within a single language, some words may be difficult to syllabify and different linguists may infer different syllabifications, since different phonological tests may yield conflicting results. In some dialects of English, for example, syllable counts for words like hire and flour are notoriously difficult to pin down.