ASCII Odd/Even Cipher

Perl, 63 62 bytes

Includes +4 for -lp

Give input on STDIN

oddeven.pl:

#!/usr/bin/perl -lp
s%.%(--$|?$n|$':$&|(ord$&&1?$n:$'))&($n=$&,~v0)%eg;y;\x7f-\xff; ;

This works as shown, but to get the claimed score this must be put in a file without final ; and newline and the \xhh escapes must be replaced by their literal values. You can do this by putting the code above in the file and running:

perl -0pi -e 's/\\x(..)/chr hex $1/eg;s/;\n$//' oddeven.pl

Python 2, 138 131 bytes

s="\0%s\0"%input();r=''
for i in range(len(s)-2):L,M,R=map(ord,s[i:i+3]);a=i%2and[R,L][M%2]|M or L|R;r+=chr(a*(a<127)or 32)
print r

Try it online (contains all test cases)

Less golfed:

def f(s):
    s="\0%s\0"%s
    r=''
    for i in range(1,len(s)-1):
        if i%2: # even (parity is changed by adding \x00 to the front)
            a=ord(s[i-1]) | ord(s[i+1])
        else:   # odd
            a=ord(s[i])
            if a%2: # odd
                a|=ord(s[i-1])
            else:   # even
                a|=ord(s[i+1])
        r+=chr(a if a<127 else 32)
    print r

Try it online (ungolfed)

I add \x00 to both sides of the string so that I don't have to worry about that during bitwise or-ing. I loop along the string's original characters, doing bitwise operations and adding them to the result, following the rules for parity.

Mathematica, 152 bytes

FromCharacterCode[BitOr@@Which[OddQ@Max@#2,#~Drop~{2},OddQ@#[[2]],Most@#,True,Rest@#]/._?(#>126&)->32&~MapIndexed~Partition[ToCharacterCode@#,3,1,2,0]]&

Explanation

ToCharacterCode@#

Converts string to ASCII codes

Partition[...,3,1,2,0]

Partitions the ASCII codes to length 3, offset 1 partitions, with padded 0s.

...~MapIndexed~...

Applys a function for each partition.

Which[...]

If...else if... else in Mathematica.

OddQ@Max@#2

Checks whether the index (#2) is odd. (Max is for flattening); since Mathematica index starts at 1, I used OddQ here, not EvenQ

Drop[#,{2}]

Takes the ASCII codes of the left and right neighbors.

OddQ@#[[2]]

Checks whether the ASCII code of the corresponding character is odd.

Most@#

Takes the ASCII codes of the character and the left neighbor.

Rest@#

Takes the ASCII codes of the character and the right neighbor.

BitOr

Applys or-operation.

/._?(#>126&)->32

Replaces all numbers greater than 126 with 32 (space).

FromCharacterCode

Converts ASCII code back to characters and join them.