Bottom one looks like a standard mono 6.35mm jack. Top one looks thinner and not something I've seen before. Possibly a specific connector for aircraft.
Found a Bose aviation headset which appears to have the same jack and it seems to listed as .206".
Searching for .206" to 3.5mm brings up some connectors, but it's not that simple apparently. Aviation headset microphones use phantom power, meaning they require additional power the sound card can't provide. There seems to be boxes that provide power while having the right connections, but they're about £40; although they seem to use a battery for the power.
If you don't need to use the microphone, then the headphone should work with the correct adaptor I would have thought. Assuming you've tried the adaptor with the 6.35mm jack, I would expect to work, even if it's stereo rather than mono. The tip is the channel and the sleeve is ground. With stereo you get an extra ring, but the tip is always a channel, whether it be left or right. If you've plugged that into the correct stereo speaker/headphone out on the sound card and got nothing, then I'm not sure why to be honest. Maybe someone else with experience of aviation headsets can offer some advice.
