I had one day, a Saturday, to visit the Pena Palace. I took an early train. I followed the Rick Steves guidebook advice, and skipped taking the public bus. I hired a Tuk-Tuk to take me to the gates, bypassing the crowds that had already formed at the bus stop near the train station. Best 5 euros I spent on that trip. I was in the first few dozen to enter the palace and got may fine photographs without having to fight the crowds. Two hours later the place was absolutely mobbed. I had lunch at the palace café, and then headed back down using the trails through the lovely forest garden that was not crowded at all.
FWIW, the ride in a very old Tuk Tuk up the bumpy road was grueling. I thought my kidneys were going to end up in my throat. Hire a more modern Tuk-Tuk that has a spring suspension if you can.
Prior to the trip, I spent a bit of time studying how to get from Lisbon to Pena Palace. Every guide book said to not drive, and to take the train from Lisbon to Sintra, then Bus 434 from downtown Sintra up the hill to Pena.
But in researching further, I saw that there were travelers who were able to drive from Lisbon, and then using Google StreetView extensively to study the route, I saw that there were parking lots on the hill, and very close to the Palace. Hence, I made the decision to drive. I suspected setting out early in the morning was the key, and that proved to be crucial.
Not knowing how bad the traffic would be around Lisbon during morning rush hour, I started out early at 6AM. At the Pena car parking lot just a couple of hundred feet from the Palace Gate, I was the 3rd car there. The parking was even free.
We hanged around for 1 hour before the Palace workers even showed up. The one thing I failed to do was to purchase online tickets, so I had to purchase tickets at the booth before entering. In just those few minutes, a few hundred people with advanced tickets already got past the Palace gate and were queuing up inside. What for? It turned out that it was to take a bus up the hill from the gate to the Palace. We skipped that and walked up the hill ourselves.
Another thing we did wrong was to linger too long on the terraces and balconies of the Palace to take photos. We did not realize that to enter the interior rooms, there was a line quickly forming, and this line cost us something like 30 minutes of waiting.
On the way out, I saw lines snaking everywhere. It would take easily 3 hours to go from the gate to the Palace interior. Driving back into town, I saw lines for the 434 bus as you mentioned.
Aye, aye, aye... All that waiting in lines would not be worth it to me, but I told my wife that if we already committed a day trip from Lisbon to Sintra we would have just toughed it out, not knowing what was ahead and how long it would take. People just waited and waited, and I told my wife I was tempted to shout to them that they should just abandon the Pena Palace and go somewhere not so mobbed. It was not really worth it. My wife agreed.
Anyway, another reason to drive the car was that so I could go elsewhere in the same day. It was tough finding a parking spot in town once we drove out of the Pena parking lot, but Monserrate a few miles out-of-town was deserted and such a joy to visit with its extensive garden. We ended the day with a drive to Boca da Roca, the westernmost point in continental Europe.
If Pena Palace was that crowded in late April, I would have to be paid big bucks to visit it in June-August. How big? It has got to be many tens of thousand. Even then, I may not do it.
PS. By the way, April 25th was a big holiday in Portugal, the so-called Carnation Revolution. And May 1st was Labor Day. Perhaps many Portuguese took this occasion for an early vacation, hence the foot traffic was so bad. Would it die out the next week? I would not know.
PPS. On this trip, we never encountered crowds as those at Sintra again, until the end of the trip when we got to Barcelona. This being the 2nd time we were in Barcelona, I don't think we ever want to go back there. Even Barcelona residents had enough with tourists, and wanted some measures to limit them.