No hat. Hair parted, brushed, oiled, under a green-lined parasol held in a big white hand.
In your example, neither of the two constructions closed by a period is a sentence, i.e., an independent clause. No hat is a noun phrase. The hat isn’t doing or being anything and nothing is explicitly being done to it: there is no verb. If it did, it would look something like this:
His hat was of black straw, except on duty in a procession, when he wore no hat. — Rosamund Cuthberson, 1837.
In the independent clause, hat is the subject of the verb was, while in the dependent when-clause, hat is the object of the verb wore, which agrees with the subject he. Both verbs are couched in the standard narrative tense: simple past.
Verbs that agree with a subject in person (I, you, she, we they…) and number (singular or plural) and are marked for tense are called finite verbs.
Hair parted, brushed, oiled
Are parted, brushed, and oiled finit verbs with hair as the subject? The hair, however, didn’t do the parting, brushing, or oiling. A finite verb is missing. Supply one and it will look something like this:
His hair was oiled and neatly parted, and his white suit had been brushed clean with immaculate care. — “Migration 17.8,”Worm: A complete web serial.
The verbs was and had are finite, agreeing with their subjects hair and suit in a passive construction. Oiled, parted, been and brushed, are non-finite verbs, specifically participles. Although we call these forms past participles to distinguish from present participles (parting, brushing, oiling) or perfect participles (having been parted/brushed/oiled), they are past, present, or perfect only in form. Only finite verbs have tense.
In your example, parted, brushed, oiled and held are also past partiples. As non-finite verb forms, they have no tense.
A sentence two paragraphs before your example shows a similar — but not identical — construction using participles:
Brought from all the recesses of the coast in all the legality of time contracts, lost in uncongenial surroundings, fed on unfamiliar food, they sickened, became inefficient, and were then allowed to crawl away and rest.
Brought, lost, and fed all function like adjectives of they. The name “participle” means that these verb forms do not give up all their characteristics as verbs — no tense, but they can take subjects — but participate in the world of nouns, adjectives, and adverbs.
While these participles have no subjects unambiguously modifying they, the phrase hair parted, brushed, oiled actually does. Compare this sentence:
Look, it is my Yusif coming, his shirt starched and white, his hair oiled down, bringing his schoolwork to his Ma. — Marguerite Thoburn Watkins, Patterns in Henna, 2009, p. 19.
While bringing his schoolwork to his Ma is a garden variety participle, in this case a present participle, the constructions with subjects shirt and hair, describe Yusif as he is approaching. Another, more common way of expressing the same thing is by using the preposition with:
_With his hair oiled and his dark glasses in place, he is no longer the charming Neo-phyte of the first movie. — Review: The Matrix Reloaded, 14 May 2003.
Grammatically, this type of construction is termed an absolute construction, a free-standing element known in traditional grammar as a nominative absolute. It does not modify a single element in an independent clause, but the clause itself. Merely because Conrad chose to make the construction truly free-standing by ending it with a period does not change the basic structure.
under a green-lined parasol held in a big white hand
Again, held is a past participle. It is not used as an absolute, but modifying parasol. If it were a finite verb marked for tense, it would look like this:
In her right hand she held a parasol. A yellow parasol. — Uday Prakash, The Girl with the Golden Parasol, 2013, p. 33.
But what about the entire prepositional phrase? I suppose one could parse it as an odd parallel to the hat, and of course if the accountant is standing under a parasol, then so is his hair. I would suggest, however, that the phrase is another example of an absolute construction. After all, the phrase ends a rather lengthy description of the accountant‘s unusual garb:
I saw a high starched collar, white cuffs, a light alpaca jacket, snowy trousers, a clean necktie, and varnished boots. No hat. Hair parted, brushed, oiled, under a green-lined parasol held in a big white hand. He was amazing, and had a penholder behind his ear.
The prepositional phrase is also a free-standing element which, like Conrad‘s period, brings the long description — except for the penholder detail — to a close. The reader should be able to imagine the accountant under the green-lined parasol, in his odd ensemble and immaculately groomed hair, as a complete figure as Conrad goes on to talk about what it means to dress like this in the Congo.