First, you should keep palatal and palatalized separate. Norwegian has a palatalized fricative [ç] but doesn't have palatalized consonants, likewise Sanskrit, likewise, Hungarian and North Saami have palatals [c͡ç ɟ͡ʝ] or simply [c ɟ] but not palatalization (on labials, alveolars) – more on Saami belo. On the other hand, Russian hand Irish have a robust contrast between plain and palatalized consonants across places of articulation. It can be hard to tell if a sound is best represented as primitive palatal [c ɟ] vs. a palatalized velar [kʲ gʲ], and any proposal for a language as having only palatalization of velars (or palatalization of alveolars, though not both) can be suspected of maybe actually having palatal as an active place of articulation.
The second problem is that palatalization (and labialization) are often analytically interchangeable with consonant plus glide analysis. Hence Mandarin might have palatalized labials and linguals, or it might have C+glide onsets. The evidence for one analysis vs. the other typically involves a theoretical decision about the advisability of adding more phonemes vs. allowing more (any) consonant cluster.
If you are looking for languages with palatal as a place of articulation (but excluding the glide [j] which is real common), there are some: much of Nilotic, Akan, Basque, many Uralic languages. Languages with palatalization at a single place or articulation would probably have to be labial, to persuasively exclude a "palatal as primary place" analysis. UPSID reports the results of an old project on accumulating segment inventories, from which you can find languages said to have just palatalized labials. It reports Angas as having only palatalized labials: but, that is a matter of analysis. UPSID reports an analysis from a paper by Burquest with palatalized labials – which he treats as C+glide clusters in his dissertation. A good reason to go with the glide analysis is that palatalized (and labialized) consonants cannot appear syllable finally. I would say there is not any clear example attested of just a single place of articulation being palatalized.