The DLNA protocol is only certified to run on nnnn-Base-T ethernet and Wi-fi. It's not designed to work over wide area networks (WANs), although it might. The virtual link between your server in your house and your parents house is over a WAN. You don't say how far apart you both live or if you have the same internet service provider. At a minimum I would say the the packets, one-way, go through at least 8-10 switches and routers, point-to-point. Traffic can queue up at each switch/router interface. Or get "dropped".
If you know the basic network utilities "ping" and "traceroute" you might be able to see how many hops your traffic takes (traceroute) , and the average latency (ping) for a packet to arrive. Video quality of Service (QoS) is sensitive to bandwidth, latency, and jitter on each link of your network path. In addition DLNA uses the UDP protocol which is not guaranteed delivery. The UDP packets can be dropped and lost forever. TCP/IP is guaranteed to arrive but not necessarily in a time frame that Plex error correction can handle.
I suspect your problem is not in your equipment (server or laptop) or Wi-fi. It's the (internet) network. I'm just getting started with Plex. I don't see where you can easily set up "pre-buffering" which is what you need.
You Tube, Hulu, Vimeo, NetFlix, etc have special network engineering done by the ISP's to deliver their traffic. Your set up is the equivalent of sending Plex media streams in packets labeled "email". So you need to buffer 5-10 minutes of movie (at the client) before you start watching, if Plex can do that. Start with 480P streams first.