With regards the question in the title, in both subs you need either

my ($hash_ref) = @_;
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or

my $hash_ref = shift;
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@_ is an array.

Update
Just to note your snippet doesn't compile

Not enough arguments for connect at C:\perm\dev\monk.pl line 15, near 
+"$handle)"
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connect is a perl function and after changing the name to my_connect we then get

Can't use string ("1") as a HASH ref while "strict refs" in use at C:\
+perm\dev\monk.pl line 16.
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Line 16 is in sub a so for me the first connect doesn't work fine (because I'm using strict and warnings). What the error is telling you is that my $hash_ref = @_; sees the array in scalar context and returns the length of the array i.e. 1.

This compiles and runs but obviously I don't know what your connect() or makeDBconnection() does.

#!/usr/bin/perl

use warnings;
use strict;

 my %parameterhash;
 my $hashref = \%parameterhash;
 a($hashref);
 b($hashref);

sub a{
 my $hash_ref = shift;
 my $dbconnection= makeDBconnection();
 my $handle = \$dbconnection;
 my_connect($handle);             # line 15
 $hash_ref->{dbhandle} = $handle; # line 16
}

sub b{
 my $hash_ref = shift;
 my $hand = $hash_ref->{dbhandle};
 my_connect($hand);
}

sub my_connect {
}

sub makeDBconnection{
}
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