I moved out of Parkchester in 2001 to go to Atlanta, after living there for 30 years. I grew up there and attended PS 106 and 127. I must say that the neighborhood changed drastically over time, and I was even robbed once, but I believe that you can be safe there if you exercise adequate levels of caution and awareness. It's much more "urban" than it used to be which stems from the fact that back in the 70's and 80's when I was growing up there, it was a very beautiful community and the management strove to maintain its image as one of the best neighborhoods in the Bronx. They did background checks on every applicant, including home visits, etc. Then, they went Condo. When that happened there were people buying up the apartments and subletting them to anyone that had cash. The neighborhood took a drastic turn for the worse as anyone was able to live in Parkchester if they found the right sublessor. And the problem worsened still, when the Condo venture was not successful and Parkchester decided to once again offer rentals. At this time they decided to accept Section 8 and the neighborhood changed yet again. It's a very different Parkchester than I used to know. However, if you live around the Oval it's usually very well maintained and pretty secure. Buildings like 1, 5, 7, 99, 14 Metropolitan Oval are close to all shopping in the area and a short walk to the train station. Those buildings have always been notoriously well-maintained and cared for. I've lived all over Parkchester, in 14 and 18 Met Oval, in 1521 Unionport Rd (my favorite building); in 1604 Met Ave., in 1594 Met. Ave, and as I said, I was there for 30 years. I have no idea where the "front" or "back" of Parkchester is, as it has always been set up in quadrants, North, East, South and West. There are areas in all of the quadrants that are better taken care of and less hangout spots, than others.
Also, be wary of realtors who call a lot of places Parkchester that really are not - but are just apartments in Parkchester's outskirts. That is definitely another consideration. Remember that there is no "Parkchester area" it's either a Parkchester apartment, or it's not - and all Parkchester apartments are the buildings within the development and they all look uniform.
And finally, I would do a drive-by of the area that the realtor is suggesting one night late on the weekend, it's cold out now, so you might not have a good idea, but drive through and see where it's quiet, where you see people hanging out. I can tell you that 1594 Met. Ave. and that area is awful. People are usually on the stoop and milling about.
Overall, I don't think it's anymore safe or dangerous than any other areas in the Bronx. But you definitely have to use the usual levels of common sense and precaution.