The Czech Consulate General in New York has a page about this. Presumably a similar situation would prevail at other Czech consulates, so this answer should help even if you do not reside in its territory.
The page notes that if you are not "a relative" you can "enclose an explanation letter why do you need the duplicate of the birth [or marriage] certificate." So it's possible that they'd give you a copy anyway, but, to increase your chances, you might want to include a letter explaining that you need the document to demonstrate your own Czech nationality. You can also include a copy of your birth certificate as evidence of your relationship.
The page links to the forms you have to submit with the request for the certificate. They reflect the same possibility, slightly more specifically. The birth certificate application says
6. V případě, že nejste v tabulce, uveďte vztah k dítěti nebo jiný právní zájem:
6. In case you are not the person in a table, relation must be stated or any other legal interest:
As far as I can tell "the person in a table" means "a person in the list of birth certificates being requested with this form." That is, you don't need to explain yourself if you are the child whose birth certificate it is, nor if you are one of the parents shown on the certificate.
The marriage certificate form similarly says
6. V případě, že nejste osobou uvedenou v tabulce, uveďte vztah k osobám nebo jiný právní zájem
(If the applicant is not listed in the table below as the Husband or Wife, please provide an explanation of the relationship or legal interest which authorizes this submission.):
This last sentence hints at the one thing that I unfortunately do not know, which is the criteria for judging whether a given explanation is legally sufficient to authorize the release of the certificate to the person who submitted the application. I guess your case is as good as it gets, but it's possible that the law prevents the certificates to be issued to you while your parents are still alive.