I primarily speak for the Golden Age of Piracy of 1690-1725, but there's a good amount that could be applied to those in the greater era of 1550-1830 (which I assume is what you mean by Age of Sail and Gupowder - another good term to use is "Age of Fighting Sail").
TL;DR - No, the kind of vessels pirates used didn't lead to Navies pushing for bigger ships. Pirates used vessels that were appropriate for their targets, which doesn't always mean the biggest and most heavily armed vessels in existence.

...ever have the resources to enter or even lead the arms race in terms of growing ship sizes?

The framing of this question is a bit odd, but I'll try my best to get the core of it:
It almost sounds like there is an assumption that pirates established themselves as an independent power and built their own ships. It's hard to answer this from that perspective, so let's start with how they got ships.

Be it at the early end or the later end of the Age of Fighting Sail, pirates of any legal or illegal shade didn't do that much "build it to rival the power of navies" building of ships. Rarely do I read of a ship being built for piracy. I want to say that Stede Bonnet is the only one (and there's always that question of if he was crazy). In the early 18th century in that climax of pirate activity from about 1713-1726 (or 1716-1723 if you really wanted to center in on large numbers of pirates) practically all ships were stolen that were in the service of pirates (and yes that includes vessels that were in legal civilian service and then had a mutiny take place on them). Now, if you're talking about those pirates that kind of fell into a more gray area of legality, or pirates that started out as legal (like as privateers), they could have built a vessel or bought one - but they wouldn't necessarily rival the big Navy vessels, they would just be built to be fast, carry many men for boarding and manning prizes, and carry enough armament to convince commerce vessels to surrender/have enough firepower to subdue said targets quickly enough in battle. Those qualities are pretty much the same qualities pirates wanted in their vessels too.

Let's reflect more on the type of vessels pirates preferred. Normally a pirate's targets are vessels with valuables. Many of the prizes pirates took over this period were simple merchant ships carrying bulk cargo of some kind. In particular times and places you do get more valuable prizes - in particular the Indian Ocean. Usually, pirates wanted a vessel that could carry enough men and firepower to convince a target to surrender or use said men and firepower quickly and well enough overwhelm and defeat said target. Speed was also valuable since you can't capture a target if you can't catch up with it, plus sometimes you'll want to run from Navy vessels. For Atlantic ocean targets, it didn't always require a humongous vessel to do this - since it was common to encounter a merchant vessel with a dozen or less crew on it. A sloop with 8 to 12 guns and 40-80 men often did the trick for this. The sloop like this could equal or outdo many of the typical merchants of the era in armory, definitely in crew numbers, and be faster. Small vessels also had a better chance of having a shallower draft and therefore getting to shallower waters and possibly escaping larger vessels pursuing them (shallow draft - meaning less of the vessel below said vessel's waterline).

A number of crews did desire a ship of force though, and often resulted in pirate ships with 20-40 guns. Slave ships often made the best conversions to ships of force. Slave ships were made to be fast since the more time at sea meant the more slaves would die at sea (so slave ships would want to make the passage from Africa to the Caribbean as quickly as possible for maximum profit). They also had to have capacity for people since they were carrying many as cargo and as crew (slave ships had larger crews to help control the slaves, defend against slave insurrection and other ships, and make up for any losses to disease in the crew later on). They often had good sized armaments for defense and for repelling slave insurrections onboard. It also helps that many slave ships were former privateer vessels (as mentioned before, privateers were often picked/designed for speed and holding a lot of men). Two of the most famous pirate ships, the La Concorde (later renamed the Queen Anne's Revenge) and the Whydah were both ships of force and former slave ships. The largest ships of force for pirates during the Golden Age had around 40-50 guns (only one pirate ship reached 50 guns, that was an anomaly, the rest seldom went that far into the 40s for guns), and there weren't that many pirates that got that far. When pirates obtained ships that large, they often had crews of well over a hundred and up around 200 men each. When crews become this large, the issues of feeding them and splitting profits became an issue. It takes a lot of food to feed that many men and profits split between 200 men will be smaller than split between 40 or 80 men. There is a pattern that a lot of the bigger pirate ships of forces tended to sail onto the African Coast and/or to the Indian Ocean during the 1690-1725 period. The best conclusion I came up with is that to go for a big target vessel in the Indian Ocean that has a lot of valuables beyond bulk goods (that often had more men and guns), pirates desired to have/thought it good to have a larger vessel to contend with them.

Now, back to the arms race bit. I established above the kind of ships they would have used. Now, did this concern European powers enough to lead to a growth in ships? I've not seen evidence that indicates ships armament and size grew because of pirate activity. With pirate ships only occasionally getting to around 40 guns (and there's a good chance many of those guns weren't the kind of heavier guns desired for ship to ship combat, but rather lighter guns that did better service in dispatching enemy crew members), Navies simply dispatched frigates at the biggest to pursue pirates (or 5th Rates in terms of the way the British rated vessel size/strength). Pirates of European origins that sailed the Atlantic and Indian Ocean during the Age of Fighting Sail never seemed to obtain anything close to top Navy ships, in particular the Ships of the Line (in other words, the big Navy warships that fought in the big pitched fleet battles). Those vessels usually had at least 60 guns and went to over 100 guns. While those vessels with guns numbering in the 50s (the Fourth Rates) still had a place in "the Line" during the Golden Age of Piracy, pirates didn't even make it to that really. So pirates didn't spur on growing ship size. If you're wondering what did - the short answer is look towards the development of ships as gun platforms, the growth of battle fleets, the increase in time that vessels spent at sea growing, and the increased importance to maritime countries to defend seagoing trade that spread across the world.

Hope that answers your question. For sources, I would say this question leads more into a pirate tactics and strategies discussion, so the best and more current secondary source on that would be Benerson Little's Sea Rover's Practice. I find myself linking that book a lot to people here on reddit. Then again, most of the pirate questions we get here seem to relate back to tactics these days, and no other book is out there is equivalent to Little's work. Also, it's not like someone 300 years ago decided to write a pirate tactic manual - it has to be pieced together from numerous, various, and different sources.