This question requires identifying the grammatically correct sentence among the given options. We need to carefully examine the rules of possessive nouns and subject-verb agreement to determine the correct choice.

Grammar: Correct Possessive Use for 'Mother's'

The core issue in some options relates to the correct formation of the possessive case. When showing ownership for a singular noun like 'mother', we add an apostrophe followed by an 's' ($'s$). For example, the car belonging to the mother is $mother's car$.

• The form $mothers'$ indicates possession by multiple mothers (plural possessive).
• The form $mothers$ (without an apostrophe) is simply the plural noun.

Option 1 uses $My mother's official car$, correctly showing possession by a single mother.

Grammar: Subject-Verb Agreement with 'Car is'

Another critical grammar rule is subject-verb agreement. The verb must agree in number with its subject. A singular subject needs a singular verb, and a plural subject needs a plural verb.

• The subject $car$ is singular.
• The corresponding singular verb is $is$.
• Therefore, the correct pairing is $car is$.
• The subject $cars$ is plural.
• The corresponding plural verb is $are$.
• An incorrect pairing would be $cars is$.

Option 1 uses $car is parked$, which correctly applies subject-verb agreement.

Analyzing Sentence Grammatical Correctness

Let's evaluate each option:

• Option 1: $My mother's official car is parked elsewhere$
  • Possessive: $mother's$ is correct for a singular mother.
  • Subject-Verb Agreement: $car is$ is correct (singular subject, singular verb).
  • Adverb: $elsewhere$ is used appropriately.
  • Conclusion: This sentence is grammatically correct.
• Option 2: $My mothers' official car is parked elsewhere$
  • Possessive: $mothers'$ implies multiple mothers. While grammatically possible if there were several mothers involved, it's less common than the singular possessive and likely not the intended meaning.
  • Subject-Verb Agreement: $car is$ is correct.
  • Conclusion: Grammatically questionable due to the plural possessive form unless a specific plural context is intended. Option 1 is preferred for standard usage.
• Option 3: $My mother's official car is parked everywhere$
  • Possessive: $mother's$ is correct.
  • Subject-Verb Agreement: $car is$ is correct.
  • Adverb: $everywhere$ changes the meaning significantly from $elsewhere$ but is grammatically valid in itself.
  • Conclusion: This sentence is also grammatically sound, but differs in meaning from Option 1. Based on standard grammar questions focusing on structure, Option 1 is the typical correct answer pattern.
• Option 4: $My mother's official cars is parked elsewhere$
  • Possessive: $mother's$ is correct.
  • Subject-Verb Agreement: $cars is$ is incorrect. The plural subject $cars$ requires the plural verb $are$ ($cars are parked$).
  • Conclusion: This sentence contains a subject-verb agreement error.

Solution: Correct Sentence Identification

Based on the analysis, Option 1 is the most appropriate choice as it correctly uses the singular possessive form ($mother's$) and maintains correct subject-verb agreement ($car is$).

Option 4 is incorrect due to the subject-verb agreement error ($cars is$). Option 2 uses a plural possessive ($mothers'$) which is less common and likely incorrect in standard context. Option 3 is grammatically sound but uses a different adverb, making Option 1 the clearest example of correct standard grammar focusing on possession and agreement.