want to parse a string in shell script

Columnizing the output is the hard part. (I would agree with Noufal here that perl is a good way to go.) However, it is doable with other more basic tools. For example:

$ cat input
foo
is a first one
foo1
is a second one
foo3
is a third one
$ ( awk 'NR % 2' input; awk 'NR % 2 == 0' input ) | paste - - - | column -s'   ' -t
foo             foo1             foo3
is a first one  is a second one  is a third one

(Note that the -s argument to column should contain a tab between the single quotes.)

Awk is perfectly fine to do string parsing.

s='foo="is a first one" foo1="is a second one" foo3="is a third one"'
echo $s | awk 'BEGIN{
    FS="\042 "
}
{
    c=NF
    for(i=1;i<=NF;i++){
        m = split($i , a, "=")
        header[i] = a[1]
        content[i]= a[2]
    }
}
END{
    for(i=1;i<=c;i++){
        printf "%s\t", header[i]
    }
    print ""
    for(i=1;i<=c;i++){
        printf "%s\t", content[i]
    }
    print ""
}
'

Is this you wanted ?

   sed -nr 's/" /"\n/g;s/=/\n/gp' <<END  
    > foo="is a first one" foo1="is a second one" foo3="is a third one"
    > END
    foo
    "is a first one"
    foo1
    "is a second one"
    foo3
    "is a third one"

Stealing the use of column from William's answer, this accepts the OP's suggested input. It requires the values to be double-quoted as per the sample input:

s='foo="is a first one" foo1="is a second one" foo3="is a third one"'
echo "$s" | awk -F'"' '
  {
    for (i=1; i<=NF; i+=2) {
      sub(/^[[:space:]]*/, "", $i)
      sub(/=$/, "", $i)
      vars=vars $i "\t"
    }
    for (i=2; i<=NF; i+=2) {vals=vals $i "\t"}
  }
  END {print vars; print vals}
' | column -t -s $'\t'

outputs

foo             foo1             foo3
is a first one  is a second one  is a third one