Today I've emailed Aena who own the lounge. According to the oneworld alliance website, oneworld sapphire and emerald members should gain access as long as using a oneworld carrier. I've also read that Iberia closed their previously used lounge and now use this one only.
Here's a URL for the oneworld website. I can't figure out how to add a link on here!
https://www.oneworld.com/airport-lounges?location=LPA
Surely this confirms that at least oneworld sapphire and emerald tiers get access.