The answer to your question is: Yes, it is possible. But in one of threee round-about ways:
$tmp = $this->property;
return $tmp::MY_CONST;
Or, IMO, a bit more intuitive:
$class = get_class($this->property);
return $class::MY_CONST;
An alternative way of doing this would be:
$r = new RefletionClass($this->property);
return $r->getConstant('MY_CONSTANT');
As to the reason why your code generates a parser error, I'm not 100% sure. I can tell you, however, that the PHP grammar is quite complicated, and a tad messy.
I wouldn't be surprized, though, to learn that $this->foo::BAR
is an expression that trips up the parser, which explains the parse error.