I think they are referring to the Special Marriage Act. But, what they said is not entirely accurate.A notice of 30 days is issued proclaiming the intention of marriage and if nobody objects, the certificate is issued. Objection can be raised by anybody. For example, the person(s) marrying may have already been married or cheated somebody in the name of marriage or one of them might be minor and misrepresented their age or there may be some other legitimate objections. If there are no objections raised, the certificate would be issued. If there are objections, they are scrutinized and the officer in charge may decide to ignore the objection if its not a legitimate one.
In general, if both of them are adults, not having parents consent won't be a legitimate objection. The bigger problem in India for such marriages is parents making false allegations. For example, the girls parents may file a false case of abduction and forced marriage. In that case, the girls testimony is of at most importance. Often, the girl is forced to withhold her testimony using emotional blackmail or using some form of coercion and the guy gets screwed in that case. There have been many cases like this.
In case of inter religion marriages like between Hindu and Muslim, these days parents are more and more resorting to making even serious allegations like the the Muslim party having terror links or that their son/daughter was brainwashed and being recruited to join ISIS etc. This is of more impact as it would involve an extensive police investigation.
https://indiankanoon.org/doc/4234/
5. Notice of intended marriage.—When a marriage is intended to be solemnized under this Act, the parties to the marriage shall give notice thereof in writing in the form specified in the Second Schedule to the Marriage Officer of the district in which at least one of the parties to the marriage has resided for a period of not less than thirty days immediately preceding the date on which such notice is given.
6. Marriage Notice Book and publication.—
(1) The Marriage Officer shall keep all notices given under section 5 with the records of his office and shall also forthwith enter a true copy of every such notice in a book prescribed for that purpose, to be called the Marriage Notice Book, and such book shall be open for inspection at all reasonable times, without fee, by any person desirous of inspecting the same.
(2) The Marriage Officer shall cause every such notice to be published by affixing a copy thereof to some conspicuous place in his office.
(3) Where either of the parties to an intended marriage is not permanently residing within the local limits of the district of the Marriage Officer to whom the notice has been given under section 5, the Marriage Officer shall also cause a copy of such notice to be transmitted to the Marriage Officer of the district within whose limits such party is permanently residing, and that Marriage Officer shall thereupon cause a copy thereof to be affixed to some conspicuous place in his office.
7. Objection to marriage.—
(1) Any person may, before the expiration of thirty days from the date on which any such notice has been published under sub-section (2) of section 6, object to the marriage on the ground that it would contravene one or more of the conditions specified in section 4.
(2) After the expiration of thirty days from the date on which notice of an intended marriage has been published under sub-section (2) of section 6, the marriage may be solemnized, unless it has been previously objected to under sub-section (1).
(3) The nature of the objection shall be recorded in writing by the Marriage Officer in the Marriage Notice Book, be read over and explained if necessary, to the person making the objection and shall be signed by him or on his behalf.
8. Procedure on receipt of objection.—
(1) If an objection is made under section 7 to an intended marriage, the Marriage Officer shall not solemnize the marriage until he has inquired into the matter of the objection and is satisfied that it ought not to prevent the solemnization of the marriage or the objection is withdrawn by the person making it; but the Marriage Officer shall not take more than thirty days from the date of the objection for the purpose of inquiring into the matter of the objection and arriving at a decision.
(2) If the Marriage Officer upholds the objection and refuses to solemnize the marriage, either party to the intended marriage may, within a period of thirty days from the date of such refusal, prefer an appeal to the district court within the local limits of whose jurisdiction the Marriage Officer has his office, and the decision of the district court on such appeal shall be final, and the Marriage Officer shall act in conformity with the decision of the court.
