The correct options are
B I forgot my bag (and phone) in the office.
D I won a prize (Rs. 2000) in the essay competition.
The punctuation mark, parentheses ( ), is used to enclose additional or
extra information in a sentence, without influencing that sentence
grammatically (the single bracket is called parenthesis whereas in a
pair, ( ) is called parentheses). If the words enclosed in the
parentheses are omitted, then the sentence won't be affected
grammatically. From the given options, if you look at B and D you'll see that the information in parentheses is additional, "and phone" is an extra information without which the sentence is still complete. Parentheses are used to enclose an amount of money, like Rs. 2000 to provide the extra information. Hence, these sentences have used parentheses correctly, options B and D are right. In A, when the word "boots" is omitted, the sentence sounds incomplete, because the question arises 'red what?'. In C, the full stop is included inside the parenthesis, but the closing punctuation mark (full stop) should be placed outside the closing parenthesis. (When a parenthetical sentence is complete on its own, then the full stop is included before the closing parenthesis as well as after the original sentence.) Hence options A and C are incorrect.