According to ICAO Doc 4444, "climb straight ahead" means "fly runway track". I assume "straight out" and "straight ahead" to be the same.
If there happens to be another departure on the parallel runway AND the crosswind component is large enough, I would clarify with ATC to make sure we are on the same page: "Tower, confirm you want us to track the extended runway centre line".
I believe every pilot should be able to eyeball an approximate heading to track the runway centreline after take-off. It's not localizer precision that's expected here, some deviation is fine. I remember during my first PPL flight lessons in a Cessna 172 (with no track information), the instructor kept glancing backwards over his shoulder to check how well I was tracking the runway centre line on my climbout.
Just curious, what kind of aircraft do you fly that doesn't offer any way of determining your track? Even old steam-gauge aircraft nowadays tend to have a GPS unit (Garmin etc.) that will give you a track indication. You could even use ForeFlight on your phone to get a track reading.
If nonetheless you can't fly a given track, just report that to ATC: "unable due to lack of track indication, request runway heading instead".
