I suggest she's quick to cry because she's responding to a lack of attention when attention is reasonable to expect. Perhaps she is feeling unloved in general.feeling unloved. She may be crying because there's a load of unmet emotional needs. Her tears may not be just for the current situation. It may help to give her more attention when she's not crying. Paying attention to what she does other times. Praising her when she's doing ok. Give five positive comments for every negative comment.
You said her mom doesn't give in to her daughter. Could she be too cold, unintentionally separating herself from her daughter which causes her to feel unloved. I would expect a mother dealing with this all day to shut her daughter out when she cried. One can validate feelings without caving in. "I hear you, know you're upset. Please go to your room until you're able to stop crying." Said in a loving like tone of voice.
Ignoring her is not working. Time to try something else. For my granddaughter who is 5 it helped when her mother did something with only her and started praising her more. Last week they went to a movie. Her mother does one thing with her every other week. On the weeks in between Mom does something with her sister.
This has not totally stopped the tears but they've helped.
I recommend a book by Adele Farber and another women. Both are professionals working with children. The title is similar to How to Talk so Kids will listen and How to Listen so Kids will Talk. How we talk makes a difference for our children. Same with adults. We withdraw or become angry when we hear something is wrong with us. I worked best with supervisors who spent friendly time with me and not just when I needed a talking to.
Added: often an older child protects the younger child when she believes the younger child is mistreated or not getting the support the needs. The older child may feel responsible because parents have placed her in that role. Or she may be trying to protect both of them from their parent's anger. You don't know the parent's reaction when you're not around.
This is the parent's responsibility. All you can do is how to treat them when they're with you.
I'm often with my grandchildren. Both their mother and I ask if you need a hug. Hugs are common in their family. A sympathetic hug fixes many things.
For Jill it's important to accept that what her parents expect of her may not be the same you expect of her. Listen. Be careful to not say or intimate that her parent's expectations are wrong. Ask questions before giving any advice. It's important to support the parents while empathizing with their children.
One thing that sometimes helps with my grandkids is to redirect their attention. For example, when out and about to say hold my hand, let's run to the corner or help me find the carrots or a pretty leaf.
Do you see whatever they might be interested in. We can't do that now. We can do this and give choices when possible.
I try to stop an activity before the kids are tired, before they start whining or not getting along. When they don't want to go to the zoo and are having difficulty staying where you are, I say, ok, it's time to go home. Sometimes the kids then choose to go to the zoo instead of going home. Giving choices is more helpful than issuing "demands."