How to generate hierarchical result (pedigree) from sql query without CTE?

To prevent abusing your database with iterated calls, SELECT the whole table just once and let php do all of the recursive work on the result set.

AMENDMENT: Since collecting the ~100,000 rows is too much heavy lifting, here is alternative advice... Rather than making up to 31 individual trips to the database within a recursive process, I'll recommend that you build a filtered array based on up to 5 trips to the database.

The following snippet is NOT tested:

$generation = 1;
$needles = [1];
$animals = [];
while ($needles && $generation < 6) {
    $sth = $db->prepare("SELECT * FROM animals WHERE akc_reg_num IN (" . implode(',', array_fill(0, count($needles), '?')) . ")");
    $sth->execute($needles);
    if ($results = $sth->fetchAll(\PDO::FETCH_ASSOC)) {
        $needles = array_filter(array_merge(array_column($results, 'akc_parent_sire'), array_column($results, 'akc_parent_dam')));
        $animals[] = array_merge($animal, $results);
    } else {
        $needles = null;
    }
    ++$generation;
}
// $animals is ready to pass to the php recursion

From an $animals result set like this:

$animals = [
    ['id' => 1, 'akc_reg_num' => 1, 'akc_parent_sire' => 2, 'akc_parent_dam' => 3],
    ['id' => 2, 'akc_reg_num' => 2, 'akc_parent_sire' => 5, 'akc_parent_dam' => 6],
    ['id' => 3, 'akc_reg_num' => 3, 'akc_parent_sire' => 9, 'akc_parent_dam' => 0],
    ['id' => 4, 'akc_reg_num' => 5, 'akc_parent_sire' => 0, 'akc_parent_dam' => 0],
    ['id' => 5, 'akc_reg_num' => 6, 'akc_parent_sire' => 7, 'akc_parent_dam' => 8],
    ['id' => 6, 'akc_reg_num' => 7, 'akc_parent_sire' => 0, 'akc_parent_dam' => 0],
    ['id' => 7, 'akc_reg_num' => 8, 'akc_parent_sire' => 0, 'akc_parent_dam' => 0],
    ['id' => 8, 'akc_reg_num' => 9, 'akc_parent_sire' => 10, 'akc_parent_dam' => 11],
    ['id' => 9, 'akc_reg_num' => 10, 'akc_parent_sire' => 0, 'akc_parent_dam' => 0],
    ['id' => 10, 'akc_reg_num' => 11, 'akc_parent_sire' => 12, 'akc_parent_dam' => 0],
    ['id' => 11, 'akc_reg_num' => 12, 'akc_parent_sire' => 0, 'akc_parent_dam' => 0]
];

Itemized tasks:

• Search your array for the row of the targeted akc_reg_num, then remove that row from the "haystack" to prevent the possibility of infinite recursion, then break the search loop for best efficiency
• If there is no matching akc_reg_num in the haystack, return an empty array
• If there is a matching akc_reg_num, loop the haystack and recurse all found parents. I am filtering out "deadends" to keep the result array small and clean.
• If both parents are found in a given generation, break the loop to prevent needless iterations.
• The recursion should continue until the generation count exceeds 4 or there are no more parents to collect.

Code: (Demo)

function buildPedigree($haystack, $akc_reg_num, $generation = 0) {
    ++$generation;
    foreach ($haystack as $index => $row) {
        if ($row['akc_reg_num'] == $akc_reg_num) {
            $result = ['sire' => $row['akc_parent_sire'], 'dam' => $row['akc_parent_dam']];
            unset($haystack[$index]);             // reduce the haystack to improve efficiency and avoid infinite loop
            break;                                // stop searching
        }
    }
    if (!isset($result)) {
        return [];  // $akc_reg_num not found
    }

    foreach ($haystack as $row) {
        if ($row['akc_reg_num'] == $result['sire']) {
            $result['sire_parents'] = array_filter(buildPedigree($haystack, $row['akc_reg_num'], $generation));  // recurse and purge empty parent arrays
            if (array_key_exists('dam_parents', $result)) {
                break;  // both parents found in generation, stop this loop
            }
        } elseif ($row['akc_reg_num'] == $result['dam']) {
            $result['dam_parents'] = array_filter(buildPedigree($haystack, $row['akc_reg_num'], $generation));  // recurse and purge empty parent arrays
            if (array_key_exists('sire_parents', $result)) {
                break;  // both parents found in generation, stop this loop
            }
        }
    }
    return $generation <= 4 ? $result : [];
} 

var_export(buildPedigree($animals, 1));

Output:

array (
    'sire' => 2,
    'dam' => 3,
    'sire_parents' => array (
        'sire' => 5,
        'dam' => 6,
        'dam_parents' => array (
            'sire' => 7,
            'dam' => 8,
        ),
    ),
    'dam_parents' => array (
        'sire' => 9,
        'sire_parents' => array (
            'sire' => 10,
            'dam' => 11,
            'dam_parents' => array (
                'sire' => 12,
            ),
        ),
    ),
)