I think it really depends on how you define intelligence...
"Intelligence has been defined in many different ways- including, but not limited to abstract thought, understanding, self-awareness, communication, reasoning, learning, having emotional knowledge, retaining, planning, and problem solving" Source:
Intelligence - Wikipedia, the free encyclopedia
If you look at parrots, you can see them analysing things, working out solutions (hmm, how do I convince my human pet to bring me that nut?; or how do I get out of this cage?), understanding (oh, if I do this, x happens, if I do y, z happens), self-awareness (you see parrots refer to themselves, an African Grey I read about "Thumper go work?" when she saw her owner getting ready for work), communication (they can communicate in our language and their own - we only bother communicating with
our language and don't bother with theirs!), reasoning (uhh... depends on the birds, macaws are VERY demanding, lol, but African Greys can reason/be reasoned with, Alex for example "Wanna go back", "Not right now. If you stay out, you can go back after.", "Can have nut?" "Yes, you can have nut", "And shoulder?", "Yes, shoulder time too" [Conversation between Alex and Irene), learning (how can I destroy this?? how can I work out this toy?? how do I say that word?), emotional knowledge (I have no doubt about this - they can understand and feel the different degrees of emotion much like humans, happiness, sadness, excitement, depression, we see it time and time again in our companions), retaining (yup, they remember what they learn!), planning (example with Joanna and her Amazon Tiko who wanted a scratch, she told him to wait, and he looked around the room, spotted a cup with feathers, climbed up, grabbed a feather and used the shaft to scratch his head - no experimentation, just straight forward analysis and planning), and problem solving (phew - just look at how fast they can solve different problems, whether it is in nature - when is it safe to go to the clay licks? wait, or observe?, how can I get this food? [Kea's are notorious problem solvers and people are constantly trying to stop them getting into their bins!], and even in captivity with their toys and daily interactions)
I have no doubt that parrots are highly intelligent - and that we may never know what they are capable of. Simply because we try and apply "our" situations to them, rather than testing in their environment and terms. (Look at Alex for an example - yes he knew words, concepts, counting, etc - but he didn't NEED to as an evolution standpoint) I think if we look at how parrots behave in a natural and stress-free environment, we can really see their full potential.
You see palm cockatoos who tap their feet on a hollow log, it doesn't make much sound, but then some males figure out that if they use a stick, it can amplify the sound which is more likely to attract a mate. It was thought for a long time that only humans really used tools, but we've seen examples of other animals who do - and parrots are a classic example.
I don't know if you can really "rate" parrot intelligence against a human - we have enough issues trying to measure IQ between people of different nationalities and cultures. The problems you present to an individual have to be relevant - if we were to take a theoretical bird test "Find the ripest fruit in the jungle", we would fail, horribly

lol - but that's because we're not adapted to that experience. Same goes when we analyse bird intelligence and get them to take our tests "How many of x?" Birds don't need to know how many of something is - if it's fruit, they're not going to care. Half they'll eat, the other half will end up on the floor of the Amazon, or the walls of the suburban home. What they will notice though is if a flock member is missing - and that's a strong emotional tie.
I'll come back to this discussion later ^-^ It looks interesting!
Note: Sorry I don't have sources for all my examples

I'm a firm believer in giving evidence and sources where needed, but I only dropped into uni today to feed my fish and wanted to post on this topic. If you are interested in the examples I used, I can give a list of resources later - just let me know.