When is "marked" pronounced with 2 syllables?

I suspect your confusion stems from the fact that markedly is always pronounced as a three syllable word. There's no euphonious way to say it with two syllables, so it hasn't undergone the syllabic reduction common for many -ed words in Modern English.

Marked alone is one syllable except in archaic/poetic usage, just as FumbleFingers has already ably demonstrated.

Aged with one syllable seems to be limited to phrases with a number of years

• A man aged 46 was arrested yesterday.

and when referring to non-human things (aged cheese, well-aged beef, unaged wine).

Generally speaking, adjectives formed from regular participles, like wanted, believed, added, whispered, etc, follow the pronunciation rules for the {-ED} past tense morpheme. {-ED} is pronounced

• /-əd/ after dental stops /t/ and /d/ (because it's impossible to say final /-td/ or /-dd/)
• /-t/ after other voiceless sounds /p k f θ s ʃ/
• /-d/ elsewhere, including after vowels

In the case of aged, the final sound is voiced, and therefore /-d/ is possible, but it's an affricate /dʒ/ formed from a dental stop, so the epenthetic shwa is possible, and occurs in some circumstances, as FF notes in his answer.

In the case of clothed (clothed all in white), either /-d/ or /-əd/ is possible because /ð/ is voiced but also dental (final /-ðd/ is hard to pronounce), though the two syllable version is usually marked with an accent to distinguish it (clothéd all in white). See Pratchett & Gaiman's Good Omens for some examples.

In the case of marked, I think Jonathan has it right -- markedly gets the epenthetic shwa to separate the /-ktl-/ cluster, with the /t/ voiced to /d/ by the preceding vowel.

The online Merriam-Webster dictionary gives the two-syllable pronunciation in audio. This pronunciation goes with the second definition:

2 or ˈmär-kəd : having a distinctive or emphasized character ⟨has a marked drawl⟩

(for some reason, this pronunciation doesn't show up on my screen, even though I was able to paste it into this window.)

I definitely remember this pronunciation being used for things like "a markèd limp" when I was younger (maybe 40-50 years ago), but it was rare even then and I would say that this pronunciation is almost obsolete now, even for this meaning, and that there is no reason for you to use it.

As for aged, I think the difference is that the one-syllable (aged cheese) has had a process worked on it, and the two syllable (aged parent) has simply gotten older. In the case of the map, if I heard it with one syllable I would think someone had intentionally made it look older, and if I heard with two syllables I would think it was just an old map.

I believe, contrary to the above, that the two-syllable pronunciation is quite common in descriptive science in general and, in particular, medical enunciations of symptoms or side-effects of a condition or medication. E.g., "topical application of X is known to cause 'mar-ked' erythema immediately surrounding the area of application".