Can a planet form before the parent star ignites?

This should be physically possible, it even might happen in the universe right now.

The creation of a stellar system starts with a cloud of matter. This cloud will collapse and form some kind of thing we later might call a star. You might also call this a planet - if the mass is not enough to create a pressure which is high enough to make an initial fusion, you will have a brown dwarf. Also Jupiter is a planet which is a "too light" star.

In my opinion, nothing speaks against the possibility, that maybe two "clusters" forming at the same time. Or a smaller cluster forms first and a bigger - which becomes the star later - forms second.

The point here is more, what you call a planet? Is the sun a planet? Would be Jupiter a planet, if it shines like a star? Maybe you can call the sun a planet, if it rotates around some bigger mass? There might be some sort of definition to be done for this, I think.

To the best of my knowledge, the current thinking is that some large chunk of matter starts sweeping up matter in the circumstellar disk during/just after the protostellar phase of the star (e.g., T-Tauri stars) and forms gas-giant planets. The origin of smaller, rocky planets are still debated, but I think most astronomers believe them to be created after the circumstellar disk has been cleared.

The answer must surely be yes; according to our current theoretical ideas and observations of circumstellar disk lifetimes, planets must be able to form prior to the ignition of the main phase of hydrogen burning on the main sequence.

The Details:

I take your question to mean planet formation prior to the fusion of hydrogen into helium, not the very brief phase during which the star's primordial deuterium is burned, which for a star like the Sun happens within the first million years and certainly will take place before planets can fully form.

Planets form in a disk of circumstellar material around their parent protostars. The "core-accretion" model of giant planet formation suggests that it takes only 5-10 million years to form a giant planet in this disk. The main competing model (thermal instability in the disk) suggests an even more rapid formation timescale. Stars of less than a solar mass take longer than this to contract sufficiently that their cores reach the necessary temperatures for hydrogen ignition. To put some numbers on this: according to the pre-main sequence models of Siess, Dufour & Forestini (2000), it takes 25 million years for a solar mass star to produce an appreciable fraction of its luminosity from hydrogen burning. This timescale gets longer for lower mass stars.

Statistically speaking, we know that most stars lose their disks within about 5 million years (see for example Hillenbrand 2008). They are essentially gone by an age of 10 million years, so all giant planet formation must have occurred by then. However, we also now know thanks to extensive surveys for exoplanets that a large fraction of stars have gas giant planets. So most stars must form planets within about 5 million years and this is definitely not enough time for H-burning to start in a star of $<1.5M_{\odot}$. The same argument probably does not apply to small rocky planets, which probably take a bit longer to reach their final configuration (maybe as long as 100 million years in our solar system), though large planetesimals should be present after only a million years.