It seems that it can be analyzed using parse_str(), but not without some conversion:
$str = '["Day1"]["Morning"]=mutton&["Day1"]["Evening"]=Juice&["Day2"]["Morning"]=burger&["Day2"]["Evening"]=pizza';
parse_str(preg_replace('/(?<=^|&)/', 'x', str_replace('"', '', $str)), $a);
var_dump($a['x']);
, x, parse_str(). , preg_replace(), :
x[Day1][Morning]=mutton&x[Day1][Evening]=Juice&x[Day2][Morning]=burger&x[Day2][Evening]=pizza
x.