(1) May I give you my opinion?
(2) I shall be using traditional names for the parts of speech. The 8 parts taught in high school: noun/pronoun/adjective/verb/adverb/preposition/conjunction/interjection.
(3) The American Red Cross = noun (proper)/ subject of the sentence
founded by Clara Barton:
(a) founded = participle that modifies "The American Red Cross."
(b) by Clara Barton = prepositional phrase modifying "founded"
(i) by = preposition
(ii) Clara Barton = noun (proper) that is object of "by."
helped = verb
people = noun/ direct object in the sentence.
in times = prepositional phrase that modifies "helped."
of disaster = prepositional phrase that modifies "times."
Thank you for your interesting sentence. Ms. Barton was, indeed, a great lady. And let's remember Mr. Jean Henri Dunant, a Swiss gentleman, who started the Red Cross.
Have a nice day!