Parliamentary sovereignty means that Parliament has
the right to make or unmake any law whatever: and, further, that no person or body is recognised by the law ... as having a right to override or set aside the legislation of Parliament.
– A. V. Dicey, via Wikipedia
The idea that Parliament is sovereign is in contrast to, for example, the US, where the role and powers of Congress are explicitly limited: Article I Section 8 lists the powers that Congress does have, while the 10th Amendment makes it clear that any and all powers not delegated explicitly or implicitly to Congress or another branch of the US government "are reserved to the States respectively, or to the people." Congress has limits on its power; it's not sovereign. Parliament is not subject to any such external limits.
Composition of Parliament
However, the most important part – which is often glossed over – is that
Parliament means ... The [Queen], the House of Lords, and the House of Commons: these three bodies acting together ... constitute Parliament.
Dicey notes that "the word has often a different sense in conversation" so perhaps the tendency to conflate Parliament and the House of Commons is not a new one. However, (though I am not a lawyer) it seems to be perfectly acceptable for the law, as passed by Parliament to affect how the component parts of Parliament behave. Parliament can regulate or empower the Queen. Parliament can regulate or empower the House of Lords. And Parliament can regulate or (as in the example of the Fixed Term Parliaments Act) empower the House of Commons.
Effect of laws on Parliamentary Procedure
Acts of Parliament can't bind Parliament, in that they can always be "unmade" (to use Dicey's wording). If Parliament were, for example, to pass a law making it a crime in Britain to speak any language other than French, there would be no way to legally entrench that law – entrenching it would "bind Parliament". Any subsequent Parliament could "unmake" it. Nevertheless, one would imagine that the Parliamentary debate that led to the law being repealed would take place en français. The law can't bind Parliament, but it can certainly affect how it does business.
A less ridiculous example of an Act of Parliament being able to affect the proceedings of Parliament is the Parliament Act 1911, which provides an alternative framework for the passing of legislation (without going into too much detail, the Act allows for (almost) any Act of Parliament to be passed without the consent of the House of Lords). Nevertheless, such laws are still legally passed by Parliament as a whole: they are not merely a resolution of the Commons. Parliament could repeal that law – it isn't entrenched. And Parliament could even use those alternative provisions to extend the scope or power of the law itself (as they did in 1949). All of this constitutes parts of Parliament acting within the constraints of -- and exercising powers granted by -- previous Acts of Parliament. But, again nothing's entrenched and Parliament is not "bound"; Parliament is still allowed to make or unmake any law.
Dissolution of Parliament / Calling an Election
As the question said,
Can the Parliament ignore the Fixed Parliament Act and announce a new general election with just a simple majority?
Parliament could indeed call an election (or do anything else) by passing a new Act of Parliament. An Act that simply called an election would contradict the FTPA, but that would be fine because any Act of Parliament supersedes any earlier Acts that conflict with it. To (mis)quote Richard Nixon, "if the Parliament does it, that means it's not illegal."
Looking at the House of Commons specifically, passing such an Act would, on the part of the House of Commons, only require a majority vote. But it would also require a majority in the House of Lords, and the Royal Assent (or would require following the alternative procedure outlined in the Parliament Acts). Again, Parliament is not the House of Commons – Parliament has three parts, and when it comes to changing the law of the land, the House of Commons can do one thing and one thing only: pass a bill and hope that -- with the consent of the Lords (more or less) and the assent of the Queen -- it becomes an Act. A simple majority vote in the House of Commons would not, in and of itself, change the law.
However the Fixed-term Parliaments Act looks like a piece of binding legislation, since it requires a two-thirds majority vote in the House of Commons to trigger a general election. [Emphasis mine]
The "2/3 vote" does look like the kind of thing you'd see in an entrenched law (see, for example, amendments to the US Constitution), but in this case the 2/3 requirement has nothing to do with changing laws – it's to do with the House of Commons triggering a procedure under the law.
Prior to the passing of the Fixed Term Parliaments Act, the right to dissolve Parliament rested with the Queen (and so, in effect, rested with the Prime Minister). The FTPA removed that power in order to prevent elections being called at a time that best suited the Prime Minister's own political interest. But, in case of emergency, an analogous power was granted to the House of Commons (not Parliament) subject to a 2/3 vote – this meant that an early election could be held if there was a broad political consensus that it was necessary. The fact that the House of Commons is part of Parliament is, for the purposes of the Act, totally irrelevant (at least practically and legally speaking, though obviously politically speaking it's highly relevant). The point is that it's a power granted by an Act of Parliament, but not granted to Parliament – from a legal or practical perspective, the power could just have easily been given to a majority of the House of Lords, or to 3/4 of an independent commission, or to 5/6 of StackExchange users. Parliament has not in any way bound itself by granting that power to the Commons: Parliament can always unmake the law and take that power away again.
Or is it binding and cannot be ignored unless it's repealed?
It is "binding", inasmuch as it's the law and "no body" (to borrow from Dicey again) can ignore it – the House of Commons included. But it doesn't "bind Parliament" because Parliament can still "unmake" it by repealing it under the normal procedure in which a majority vote of the Commons is indeed necessary but not sufficient. And rather than trying to get the Lords on board and justify a wholesale change in the law, they may be better off just trying to cobble together 2/3 of the Commons.