If you are able to use Perl, you can do it in one line like this:
perl -0777 -pe 's/my first line\nmy second line\nmy third line\nthe fourth\n5th and last\n//g' paragraph_file
the explanation is in perlrun:
The special value 00 will cause Perl to slurp files in paragraph mode. The value 0777 will cause Perl to slurp files whole because there is no legal byte with that value.
Sample input:
my first line
my second line
my third line
the fourth
5th and last
hey
my first line
my second line
my third line
the fourth
5th and last
hello
my first line
Output:
$ perl -0777 -pe 's/my first line\nmy second line\nmy third line
\nthe fourth\n5th and last\n//g' paragraph_file
hey
hello
my first line