dns for subdomain without domain

[Steve]

Is it possible to setup DNS for a subdomain without setting up dns for
the parent domain?  For instance, I want to setup just DNS (an A/MX
record) for mail.foo.com but don't want foo.com to resolve.  What
would the syntax for this be if its possible?  I want mail to work for
a subdomain but the domain not to work I guess.  If this is not
possible what are the alternatives?  Do I need to have an A record for
the parent domain?  What if I wanted other subdomains (A/CNAME
records) to work also such as booger.foo.com