If you are sure you need to be able to read/write the DS1286 outside of the equipment that uses it, a PIC programmer *WON'T* help.
An Arduino Uno or Nano has enough I/O pins to access a DS1286. It needs six address lines
(connect to Arduino A0-A5), 8 data lines
(connect the two LSBs to digital pins 8 and 9, and the rest of the data bus to digital pins 2-7, matching the data bit number to the arduino pin number mod 8 to make the sketch easier to code#), and three control lines: /CS, /OE and /WE (connect to digital pins 10-12). The easiest hardware setup would be an Arduino Nano in a solderless breadboard with the DS1286 chip. It only needs max. 15mA @5V, so you can power it from the Arduino, powered from your PC via USB. It will need a 100nF decoupling cap between its Vcc pin and GND pin.
Then you need to write a sketch to dump the memory contents. If you get stuck ask here. *DON'T* test your sketch on a good chip from your equipment, get a spare or use one that's failed and been hacked to replace the battery. During development, use 3K3 resistors in series with the data lines to prevent disaster if you accidentally enable /OE and /CE with the data bus pins set as outputs in the sketch. Once you've got that working we can help you add the ability to write data to specific registers/locations, and once that's tested and working you can risk pulling a good chip to dump it so you can program the replacements.
N.B. *ALL* N.O.S. DS1286 modules are date expired. They only are specified for a ten year operating or shelf life from date of manufacture and Maxim discontinued production back in 2006, so any old stock you can find is already at 150% of its expected lifespan and is likely to either have a dead battery or to fail within a couple of years.Re: Techman's suggestion, the battery can be approximately located with a small powerful magnet.
There's a thread on how to Dremel in to the battery connections and patch in an external CR2032 coin cell here:
https://forums.irixnet.org/thread-1370.htmlHowever I would recommend using a small cylindrical burr from a set of Dremel burr bits rather than a drill bit to grind away enough epoxy to expose the battery pins. Thinner wire would make a neater job of connecting the added battery holder. If you don't have clearance above the chip, it could be located elsewhere.
I *DON'T* recommend the approach used in the attached photos I found on Google,
but they do give you an idea of the battery and crystal locations.
# ATmega328P based Arduinos, including the Uno and Nano use digital pins 0 and 1 to communicate serially with the host PC, so you can't simply use digital pins 0-7 (AVR port D) for the data bus. The choice of pins above uses the six upper bits of port D and the lower two of port B, so you can split the data between them on write and combine them on read using direct AVR register access and C bitwise logic operators rather than having to loop through reading or writing each byte one bit at a time using the Arduino single pin access library functions digitalWrite() and digitalRead(). Similarly you'll be setting the address directly on A0-A5 by writing port C.