I can give no tips as I aspire to run that fast - all I can say is that this sort of thing generally sorts itself out the more you run and the less you think about it
Sounds to me is that what you are lacking is stamina and not speed. Endurance is the ability to run a certain distance at a slowish pace - stamina is the ability to run that distance at a certain specified pace- and speed is something that you only do for very short times/distances ( like if you "sprint" to the finish line or overtake somebody ) So - you now have enough endurance to run for 5k but you don't have the stamina to do it at the pace you want. Two things can help with this - firstly endurance and stamina are somewhat linked so long slow runs out to say 10K will help with 5K stamina. But really what you need is to do multiple repeats of shorter runs (say 500 metres with a rest inbetween) at the pace you wish to average over 5K. So try running multiples of 500 metres at say 6:30 pace. When you can complete 10 of these (should take quite a few weeks) , then do it all over again but using 1 klm repeats ( 5 of them) - when you can finish these go to 1 mile repeats ( 3 of them) - eventually you will be able to do the complete 5K at 6:30 pace -- or not!, everybody has their personal limits.
I'm not very technical but I think you are doing okay, you can't be slowing that much to be near 35mins. I always take it to mean don't head off in a Usain Bolt style sprint so that you are knackered before the end of the first Km. In races/Parkrun I've often been mowed down by people careering off and later I trot on past them thinking of the tortoise and the hare. As you build stamina it will sort itself out.
ps: I think Parkrun is the height of achievement.
Have you tried the speed podcasts and fartlek? I am a steady runner and at present sprinting is out of the question, but before Christmas I tried both and it did make a difference to my overall pace.
I’m the other way round! My first k is always my slowest. Strange isn’t it!
I always consciously start of what I thinki is slowly, only to find it was my fastest lap. But on the other hand I gain stamina as the run goes on - usually around the 20 minute mark - then feel as if I could just keep going.
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) So - you now have enough endurance to run for 5k but you don't have the stamina to do it at the pace you want. Two things can help with this - firstly endurance and stamina are somewhat linked so long slow runs out to say 10K will help with 5K stamina. But really what you need is to do multiple repeats of shorter runs (say 500 metres with a rest inbetween) at the pace you wish to average over 5K. So try running multiples of 500 metres at say 6:30 pace. When you can complete 10 of these (should take quite a few weeks) , then do it all over again but using 1 klm repeats ( 5 of them) - when you can finish these go to 1 mile repeats ( 3 of them) - eventually you will be able to do the complete 5K at 6:30 pace -- or not!, everybody has their personal limits.
Slow start, slow middle and slow finish
Speed Invervals False Start!
Another slow shuffle 
CBDB’s Slow Run Fun Plan for post 10k consolidation 
I know I'm slow but...
