There are 2 ways to get to Shetland - Plane or Ferry. Plane is more expensive but takes a lot less time. Loganair fly to Shetland from many cities in Scotland. For the ferry you have to go to Aberdeen and its an overnight ferry which does get you into Lerwick in plenty of time for the Bressay ferry although its not the same port and the Aberdeen ferry docks in the north of Lerwick. There is a bus which goes from Sumburgh airport to Lerwick but if you want to explore the island you are as well renting a car.
On parkrun day itself I would arrive at the ferry terminal in Lerwick 20 mins -25 mins before hand. The ferry was there waiting when I went. You pay when you go on the ferry. It was £2.90 I think and you only pay on the way out. They accept contactless, cash or chip and pin. I didn't need cash at all for buses either, used contactless for everything.
The ferry takes about 10 minutes and there is a muster area once you get to Bressay at the car park for the ferry terminal. From memory there are toilets here too as well as a waiting area. You can leave suitcases in the ferry waiting area at your own risk. They also have a white van which you can put your jackets/warm up gear in and they drive it to the finish. The first timer briefing and part of the main briefing also take place here before everyone walks over to the mayfield hotel where the parkrun starts.
The parkrun itself goes south about 3k. Turns round and comes back but not as far as the ferry. It finishes in the cafe car park. The cafe is run by parkrun volunteers and I think the money goes to charity. Well worth a visit. It is about 20 minute walk back from the cafe to the ferry. Or at least allow that time if you are walking. Then the ferry is at 10.30 or another at 11.30. 11.30 is not the last ferry though there are plenty more crossings throughout the day. You can take a car on the ferry but if you are just doing parkrun and not exploring its probably not worth it. I believe there was parking on the Lerwick side beside the port.