External hard drive 2 macs

Hi.

I have an external hard drive that I use with my MacBook Air. This is to store photos and videos. I can read and write to the HD absolutely fine.

When i I connect the hard drive to another MacBook Air, I can only read and cannot add additional files to the HD.

Is there any way that this can be changed to read and write? I currently don’t have access to the first MacBook. I don’t want to lose the data already on the hard drive.

Many thanks.