İ couldn't find much info on internet about solubility of glycerol in triglycerides. I suppose it is non soluble because glycerin is polar but fats aren't. But fats have glycerin side in them too, so can they interact?
Glycerol (glycerin) is polar molecule with three hydroxy groups. Fats are tri-esters of glycerol and very long-chained fatty acids, and hence relatively non polar molecules. Thus it is safe to assume that fats might not be dissolved in glycerol and vise versa. This assumption is based on the fact that glycerol is insoluble in biodiesel (Ref.1). An excerpt from Reference 1 states:
Glycerol is essentially insoluble in biodiesel so that almost all glycerol is easily removed by settling or centrifugation. Free glycerol may remain either as suspended droplets [...]
Biodiesel is long-chain fatty acid esters (mostly ethyl, methyl, or propyl esters). The US National Biodiesel Board defines "biodiesel" as a mono-alkyl ester. Thus, if glycerol did not dissolve in long-chain mono-alkyl ester, it is safe to assume that it won't dissolve in long-chain tri-ester such as fats.
OP has also shown interest in the solubility of glycerol in fatty acids, specifically 18-carbon fatty acids (omega 3). That can be done in home easily in smaller scale to find out the solubility. However it is worth noting of this report, which states that:
[..] have studied the purification of glycerol from fatty acids. For the binary systems glycerol-fatty acids, our experimental data show an increasing values of solubility with the number of double-carbon bonds, because of the polarity of this group.
- Jon Van Gerpen, Gerhard Knothe, Michael J. Haas, Alfred K. Schultz, Rajiv Banavali, Klaus-Dieter Topp, Mark T. Vandersall, "Chapter 4: Biodiesel Production," In The Biodiesel Handbook, 2nd Edition; Gerhard Knothe, Jürgen Krahl, Jon Van Gerpen, Eds.; AOCS Press: Urbana, IL, 2010, pp. 31-96 (ISBN: 978-1-893997-62-2).