Translation:My father goes to his office by train.
In headlines, words that carry grammatical meaning (such as prepositions) rather than lexical meaning are commonly left out. However, that does not make it grammatically acceptable in any other context. In everyday use. "I left it on train" is both awkward and grammatically incorrect.
Yes in formal tone your office is correct. I found this note that can help us in future when composing sentences. " The forms su and suyo are ambiguous and for clarity or emphasis sake may be replaced by the expressions de usted, de él, de ella, de ustedes, de ellas, and de ellos." What do you think?
If you're a native English speaker, please go ahead and use the present progressive whenever appropriate and tell Duolingo to accept it (via ‘Report a Problem’), to make life easier for others, especially so that users whose native language isn't English won't be mistaught that the Spanish present indicative can always be translated with the English present indicative.
Since "su" can mean "his, hers or its" more context would be needed.
In general, when there is not additional context, we should assume that the "su" in this sentence refers back to the person already mentioned.
My father goes to HIS office.
"My father goes to her office...." would be said exactly the same way. It might also be worded "Mi padre va a la oficina de ella en tren" to avoid this type of confusion.