Good news, I have successfully found a pacer!
Now, here's the story:
I went to Harrogate Station and waited for the 1415 to Leeds. Disappointingly, it was a Class 153, not a pacer.
However, that all changed while I was pulling into Leeds. I saw two types of pacer, the Class 142 and the 144. I was extremely happy. On platform 5c, there was 142 050 lingering there. I asked where it would be going, but nobody knew and one person said it wouldn't be going anywhere for an hour. That didn't matter though.
On platform 11c, the 1518 service to Manchester Victoria was waiting, and it consisted of a Class 142 and a type of Sprinter, though I don't know which type as I was in a little bit of a hurry and to me, the Pacer only mattered. I went to Bradford Interchange, but for a while, I thought I had a return to Bramley instead. Then we got the 1549 Class 158 back to Leeds.
My mum was the first person to spot that our train back to Harrogate was on Platform 5c and we soon discovered that it might be that very Pacer waiting to move, and in fact, it was! Then I paced along back to Harrogate.
At Leeds, I saw all sorts of trains from the Class 82 all the way to Class 333s, it was brilliant!
Thanks for your help,
Max