There's a database compiled by volunteers, called FreeDMB, which will eventually provide free access to details of births, marriages and deaths in England & Wales from 1837 onwards. However, as these charts show, coverage of the 1960s is still very limited:
http://www.freebmd.org.uk/progressM.shtml
If you think that it might still be worth a try, start here:
http://www.freebmd.org.uk/cgi/search.pl
Otherwise you'll have to seek a copy of the marriage certificate from the GRO:
https://www.gov.uk/research-family-history
One possible alternative though would be to check the records of the church (if appropriate) where the ceremony took place. Does a family member know it, perhaps? (Or perhaps your parent's marriage certificate ended up with one of your siblings after their deaths?)
Or simply ring the Passport Adviceline to see if it's possible to submit a valid application without knowing the date of your parent's marriage:
https://www.gov.uk/passport-advice-line